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ON THE LOCATION OF ZEROS OF THE LAPLACIAN MATCHING POLYNOMIALS OF GRAPHS

JIANG-CHAO WAN, YI WANG, ALI MOHAMMADIAN

School of Mathematical Sciences, Anhui University, Hefei 230601, Anhui, China

ABSTRACT. The Laplacian matching polynomial of a graph $G$, denoted by $\mathcal{LM}(G,x)$, is a new graph polynomial whose all roots are nonnegative real numbers. In this paper, we investigate the location of zeros of the Laplacian matching polynomials. Let $G$ be a connected graph. We show that $0$ is a root of $\mathcal{LM}(G,x)$ if and only if $G$ is a tree. We prove that the number of distinct positive zeros of $\mathcal{LM}(G,x)$ is at least equal to the length of the longest path in $G$. It is also established that the zeros of $\mathcal{LM}(G,x)$ and $\mathcal{LM}(G-e,x)$ interlace for each edge $e$ of $G$. Using the path-tree of $G$, we present a linear algebraic approach to investigate the largest zero of $\mathcal{LM}(G,x)$ and particularly to give tight upper and lower bounds on it.

1. INTRODUCTION

The graph polynomials, such as the characteristic polynomial, the chromatic polynomial, the independence polynomial, the matching polynomial, and many others, are widely studied and play important roles in applications of graphs in several diverse fields. The location of zeros of graph polynomials is a main topic in algebraic combinatorics and can be used to describe some structures and parameters of graphs. In this paper, we focus on the location of zeros of the Laplacian matching polynomials of graphs. For more results on the location of zeros of graph polynomials, we refer to [9].

Throughout this paper, all graphs are assumed to be finite, undirected, and without loops or multiple edges. Let $G$ be a graph. We denote the vertex set of $G$ by $V(G)$ and the edge set of $G$ by $E(G)$. Let $M$ be a subset of $E(G)$. We denote by $V(M)$ the set of vertices of $G$ each of which is an endpoint of one of the edges in $M$. If no two distinct edges in $M$ share a common endpoint, then $M$ is called a matching of $G$. The set of matchings of $G$ is denoted by $\mathcal{M}(G)$. A matching $M \in \mathcal{M}(G)$ is said to be perfect if $V(M) = V(G)$. The matching polynomial of $G$ is

$$\mathcal{M}(G,x)=\sum _{M\in \mathcal{M}(G)}(-1{)}^{\mathrm{\mid }M\mathrm{\mid }}{x}^{\mathrm{\mid }V(G)\setminus V(M)\mathrm{\mid }}\backslash mathcal\{M\}(G,x)\; =\; \backslash sum\_\{M\; \backslash in\; \backslash mathcal\{M\}(G)\}\; (-1)^\{|M|\}\; x^\{|V(G)\; \backslash setminus\; V(M)|\}$$

which was formally defined by Heilmann and Lieb [7] in studying statistical physics, although it has appeared independently in several different contexts.

The matching polynomial is a fascinating mathematical object and attracts considerable attention of researchers. For an instance, by studying the multiplicity of zeros of the matching polynomials, Chen and Ku [8] gave a generalization of the Gallai–Edmonds theorem which is a

2020 Mathematics Subject Classification. Primary: 05C31, 05C70. Secondary: 05C05, 05C50, 12D10. Key words and phrases. Graph polynomial, Matching, Subdivision of graphs, Zeros of polynomials. Email adress:wanjc@stu.ahu.edu.cn (J.-C. Wan), wangy@ahu.edu.cn (Y. Wang, corresponding author), ali.m@ahu.edu.cn (A. Mohammadian). Funding. The research of the second author is supported by the National Natural Science Foundation of China with grant numbers 11771016 and 11871073. The research of the third author is supported by the Natural Science Foundation of Anhui Province with grant number 2008085MA03. ---PAGE_BREAK---

structure theorem in classical graph theory. For another instance, using a well known upper bound on zeros of the matching polynomials, Marcus, Spielman, and Srivastava [10] established that infinitely many bipartite Ramanujan graphs exist. Some earlier facts on the matching polynomials can be found in [4].

We want to summarize here some basic features of the zeros of the matching polynomial. For this, let us first introduce some more notations and terminology which we need. For a vertex $v$ of a graph $G$, we denote by $N_G(v)$ the set of all vertices of $G$ adjacent to $v$. The degree of $v$ is defined as $|N_G(v)|$ and is denoted by $d_G(v)$. The maximum degree and the minimum degree of the vertices of $G$ are denoted by $\Delta(G)$ and $\delta(G)$, respectively. For a subset $W$ of $V(G)$, we shall use $G[W]$ to denote the induced subgraph of $G$ induced by $W$ and we simply use $G-W$ instead of $G[V(G)\setminus W]$. Also, for a vertex $v$ of $G$, we simply write $G-v$ for $G - {v}$. For an edge $e$ of $G$, we denote by $G-e$ the subgraph of $G$ obtained by deleting the edge $e$.

Let $\alpha_1 \le \dots \le \alpha_n$ and $\beta_1 \le \dots \le \beta_m$ be respectively the zeros of two real rooted polynomials $f$ and $g$ with $\deg f = n$ and $\deg g = m$. We say that the zeros of $f$ and $g$ interlace if either

$${\alpha }_1\le {\beta }_1\le {\alpha }_2\le {\beta }_2\le \dots \backslash alpha\_1\; \backslash le\; \backslash beta\_1\; \backslash le\; \backslash alpha\_2\; \backslash le\; \backslash beta\_2\; \backslash le\; \backslash dots$$

or

$${\beta }_1\le {\alpha }_1\le {\beta }_2\le {\alpha }_2\le \dots \backslash beta\_1\; \backslash le\; \backslash alpha\_1\; \backslash le\; \backslash beta\_2\; \backslash le\; \backslash alpha\_2\; \backslash le\; \backslash dots$$

in which case one clearly must have $|n-m| \le 1$. We adopt the convention that the zeros of any polynomial of degree 0 interlace the zeros of any other polynomial.

For any connected graph $G$, the assertions given in (1.1)-(1.3) are known.

(1.1) All the roots of $\mathcal{M}(G, x)$ are real. Moreover, if $\Delta(G) \ge 2$, then the zeros of $\mathcal{M}(G, x)$ lie in the interval $(-2\sqrt{\Delta(G)-1}, 2\sqrt{\Delta(G)-1})$ [7].

(1.2) The number of distinct roots of $\mathcal{M}(G, x)$ is at least equal to $\ell(G)+1$, where $\ell(G)$ is the length of the longest path in $G$ [5].

For each vertex $v \in V(G)$, the zeros of $\mathcal{M}(G-v, x)$ interlace the zeros of $\mathcal{M}(G, x)$.

(1.3) In addition, the largest zero of $\mathcal{M}(G, x)$ has the multiplicity 1 and is greater than the largest zero of $\mathcal{M}(G-v, x)$ [6].

Recently, Mohammadian [11] introduced a new graph polynomial that is called the Laplacian matching polynomial and is defined for a graph $G$ as

$$(1.4)\mathcal{L}\mathcal{M}(G,x)=\sum _{M\in \mathcal{M}(G)}(-1{)}^{\mathrm{\mid }M\mathrm{\mid }}(\prod _{v\in V(G)\setminus V(M)}(x-d_G(v)))\mathrm{.}(1.4)\; \backslash qquad\; \backslash mathcal\{LM\}(G,x)\; =\; \backslash sum\_\{M\; \backslash in\; \backslash mathcal\{M\}(G)\}\; (-1)^\{|M|\}\; \backslash left(\; \backslash prod\_\{v\; \backslash in\; V(G)\; \backslash setminus\; V(M)\}\; (x\; -\; d\_G(v))\; \backslash right).$$

Mohammadian proved that all roots of $\mathcal{LM}(G, x)$ are real and nonnegative, and moreover, if $\Delta(G) \ge 2$, then the zeros of $\mathcal{LM}(G, x)$ lie in the interval $[0, \Delta(G) + 2\sqrt{\Delta(G)-1})$. By observing this interval, it is natural to ask: What is the sufficient and necessary condition for 0 to be a root of $\mathcal{LM}(G, x)$? More generally, as a new real rooted graph polynomial, it is natural to investigate the properties of zeros such as the interlacing of zeros, the upper and lower bounds of the largest zero, the maximum multiplicity of zeros, and the number of distinct zeros. In this paper, we mainly prove that the assertions given in (1.5)-(1.7) hold for any connected graph $G$, letting $\ell(G)$ be the length of the longest path in $G$.

If $\Delta(G) \ge 2$, then the zeros of $\mathcal{LM}(G, x)$ are contained in the interval $[0, \Delta(G) + 2\sqrt{\Delta(G)-1}\cos\frac{\pi}{2\ell(G)+2}]$, and in addition, the upper bound of the interval is a zero of $\mathcal{LM}(G, x)$ if and only if $G$ is a cycle.

(1.5) ---PAGE_BREAK---

$$(1.6)\text{The number of distinct positive roots of }\mathcal{L}\mathcal{M}(G,x)\text{ is at least equal to }\mathrm{\ell }(G)\mathrm{.}\text{ Also, if }\delta (G)\ge 2,\text{ then }\mathcal{L}\mathcal{M}(G,x)\text{ has at least }\mathrm{\ell }(G)+1\text{ distinct positive roots.}(1.6)\; \backslash quad\; \backslash text\{The\; number\; of\; distinct\; positive\; roots\; of\; \}\; \backslash mathcal\{LM\}(G,\; x)\; \backslash text\{\; is\; at\; least\; equal\; to\; \}\; \backslash ell(G).\; \backslash text\{\; Also,\; if\; \}\; \backslash delta(G)\; \backslash ge\; 2,\; \backslash text\{\; then\; \}\; \backslash mathcal\{LM\}(G,\; x)\; \backslash text\{\; has\; at\; least\; \}\; \backslash ell(G)\; +\; 1\; \backslash text\{\; distinct\; positive\; roots.\}$$

For each edge $e \in E(G)$, the zeros of $\mathcal{L}\mathcal{M}(G,x)$ and $\mathcal{L}\mathcal{M}(G-e,x)$ interlace in the sense that, if $\alpha_1 \le \cdots \le \alpha_n$ and $\beta_1 \le \cdots \le \beta_n$ are respectively the zeros of $\mathcal{L}\mathcal{M}(G,x)$ and $\mathcal{L}\mathcal{M}(G-e,x)$ in which $n = |V(G)|$, then $\beta_1 \le \alpha_1 \le \beta_2 \le \alpha_2 \le$ $\cdots \le \beta_n \le \alpha_n$. Further, the largest zero of $\mathcal{L}\mathcal{M}(G,x)$ has the multiplicity 1 and is strictly greater than the largest zero of $\mathcal{L}\mathcal{M}(H,x)$ for any proper subgraph $H$ of $G$.

It should be mentioned that the Laplacian matching polynomial is recently studied under a different name and expression by Chen and Zhang [17].

For a graph $G$, the subdivision of $G$, denoted by $S(G)$, is the graph derived from $G$ by replac- ing every edge $e = {a,b}$ of $G$ with two edges ${a,v_e}$ and ${v_e,b}$ along with the new vertex $v_e$ corresponding to the edge $e$. We know from a result of Yan and Yeh [16] that

$$(1.8)\mathcal{M}(S(G),x)=x^{\mathrm{\mid }E(G)\mathrm{\mid }-\mathrm{\mid }V(G)\mathrm{\mid }}\mathcal{L}\mathcal{M}(G,x^2)(1.8)\; \backslash qquad\; \backslash mathcal\{M\}(S(G),x)\; =\; x^\{|E(G)|-|V(G)|\}\; \backslash mathcal\{L\}\backslash mathcal\{M\}(G,x^2)$$

for any graph $G$, which is also proved by Chen and Zhang [17] by different method. The equality (1.8) shows that the problem of the location of zeros of the Laplacian matching polynomial of a graph $G$ can be transformed into the problem that deals with the location of zeros of the matching polynomial of $S(G)$. For an instance, using (1.8) and the first statement in (1.1), it immediately follows that the zeros of $\mathcal{LM}(G,x)$ are nonnegative real numbers. The assertion (1.6) is proved in Section 2 by the subdivision of graphs.

One of the most important tools in the theory of the matching polynomial is the concept of ‘path-tree’ which is introduced by Godsil [5]. Given a graph $G$ and a vertex $u \in V(G)$, the path-tree $T(G, u)$ is the tree which has as vertices the paths in $G$ which start at $u$ where two such paths are adjacent if one is a maximal proper subpath of the other. In Section 3, we show that the path-tree is also applicable for the Laplacian matching polynomial by making some appropriate adjustments. Using this, we prove (1.5) which is a slight improvement of the second statement of Theorem 2.6 of [11]. The assertion (1.7) is proved in Section 3 by linear algebra arguments.

Let us introduce more notations and definitions before moving on to the next section. We use $\lambda(f(x))$ to denote the largest zero of a real rooted polynomial $f(x)$. For a square matrix $M$, we shall use $\varphi(M, x)$ to denote the characteristic polynomial of $M$ in the indeterminate $x$. If all the roots of $\varphi(M, x)$ are real, then its largest zero is denoted by $\lambda(M)$. For a graph $G$, the adjacency matrix of $G$, denoted by $A(G)$, is a matrix whose rows and columns are indexed by $V(G)$ and the $(u, v)$-entry is 1 if $u$ and $v$ are adjacent and 0 otherwise. Let $D(G)$ be the diagonal matrix whose rows and columns are indexed as the rows and the columns of $A(G)$ with $d_G(v)$ in the $v$th diagonal position. The matrices $L(G) = D(G) - A(G)$ and $Q(G) = D(G) + A(G)$ are respectively said to be the Laplacian matrix and the signless Laplacian matrix of $G$. It is known that $\mathcal{M}(G, x) = \varphi(A(G), x)$ if and only if $G$ is a forest [14]. In addition, it is proved that $\mathcal{LM}(G, x) = \varphi(L(G), x)$ if and only if $G$ is a forest [11]. Among other results, we present a generalization of these results in Section 2.

2. SUBDIVISION OF GRAPHS AND THE LAPLACIAN MATCHING POLYNOMIAL

In this section, we examine the location of zeros of the Laplacian matching polynomial by establishing a relation between the Laplacian matching polynomial of a graph and the matching polynomial of the subdivision of that graph. Then, by analysing the structures of the subdivision of graphs, we will prove (1.6). To begin with, we recall the multivariate matching polynomial that covers both the matching polynomial and the Laplacian matching polynomial. This multivariate graph polynomial was introduced by Heilmann and Lieb [7]. ---PAGE_BREAK---

Let $G$ be a graph and associate the vector $\mathbf{x}G = (x_v){v \in V(G)}$ with $G$ in which $x_v$ is an indeterminate corresponding to the vertex $v \in V(G)$. Notice that, for a subgraph $H$ of $G$, $\mathbf{x}_H$ is the vector that has the same coordinate as $\mathbf{x}_G$ in the positions corresponding to the vertices in $V(H)$. The multivariate matching polynomial of $G$ is defined as

$$(2.1)\mathfrak{M}(G,{\mathbf{x}}_G)=\sum _{M\in \mathcal{M}(G)}(-1{)}^{\mathrm{\mid }M\mathrm{\mid }}\left(\prod _{v\in V(G)\setminus V(M)}{x}_v\right)\mathrm{.}(2.1)\; \backslash qquad\; \backslash mathfrak\{M\}(G,\; \backslash mathbf\{x\}\_G)\; =\; \backslash sum\_\{M\; \backslash in\; \backslash mathcal\{M\}(G)\}\; (-1)^\{|M|\}\; \backslash left(\; \backslash prod\_\{v\; \backslash in\; V(G)\; \backslash setminus\; V(M)\}\; x\_v\; \backslash right).$$

Let $\mathbf{1}G$ be the all one vector of length $|V(G)|$. Also, for a subgraph $H$ of $G$, we let $\mathbf{d}{G,H} = (d_G(v))_{v \in V(H)}$. For simplicity, we write $\mathbf{d}G$ instead of $\mathbf{d}{G,G}$. We sometimes drop the subscript of the vector symbols if there is no possible confusion. It is easy to see that

$$(2.2)\mathfrak{M}(G,x{\mathbf{1}}_G)=\mathcal{M}(G,x)(2.2)\; \backslash qquad\; \backslash mathfrak\{M\}(G,\; x\backslash mathbf\{1\}\_G)\; =\; \backslash mathcal\{M\}(G,\; x)$$

and

$$(2.3)\mathfrak{M}(G,x{\mathbf{1}}_G-{\mathbf{d}}_G)=\mathcal{L}\mathcal{M}(G,x)\mathrm{.}(2.3)\; \backslash qquad\; \backslash mathfrak\{M\}(G,\; x\backslash mathbf\{1\}\_G\; -\; \backslash mathbf\{d\}\_G)\; =\; \backslash mathcal\{L\}\backslash mathcal\{M\}(G,\; x).$$

Note that

$$\mathfrak{M}(G_1\cup G_2,({\mathbf{x}}_{G_1},{\mathbf{x}}_{G_2}))=\mathfrak{M}(G_1,{\mathbf{x}}_{G_1})\mathfrak{M}(G_2,{\mathbf{x}}_{G_2}),\backslash mathfrak\{M\}(G\_1\; \backslash cup\; G\_2,\; (\backslash mathbf\{x\}\_\{G\_1\},\; \backslash mathbf\{x\}\_\{G\_2\}))\; =\; \backslash mathfrak\{M\}(G\_1,\; \backslash mathbf\{x\}\_\{G\_1\})\backslash mathfrak\{M\}(G\_2,\; \backslash mathbf\{x\}\_\{G\_2\}),$$

where $G_1 \cup G_2$ denotes the disjoint union of two graphs $G_1$ and $G_2$. So, in what follows, we often restrict our attention on connected graphs.

We need the following useful lemma in the sequel.

Lemma 2.1 (Amini [1]). Let $G$ be a graph. For any vertex $v \in V(G)$,

$$\mathfrak{M}(G,{\mathbf{x}}_G)=x_v\mathfrak{M}(G-v,{\mathbf{x}}_{G-v})-\sum _{w\in N_G(v)}\mathfrak{M}(G-v-w,{\mathbf{x}}_{G-w})\mathrm{.}\backslash mathfrak\{M\}(G,\; \backslash mathbf\{x\}\_G)\; =\; x\_v\; \backslash mathfrak\{M\}(G\; -\; v,\; \backslash mathbf\{x\}\_\{G-v\})\; -\; \backslash sum\_\{w\; \backslash in\; N\_G(v)\}\; \backslash mathfrak\{M\}(G\; -\; v\; -\; w,\; \backslash mathbf\{x\}\_\{G-w\}).$$

By combining Lemma 2.1 and (2.2), we get

$$(2.4)\mathcal{M}(G,x)=x\mathcal{M}(G-v,x)-\sum _{w\in N_G(v)}\mathcal{M}(G-v-w,x),(2.4)\; \backslash qquad\; \backslash mathcal\{M\}(G,\; x)\; =\; x\backslash mathcal\{M\}(G-v,\; x)\; -\; \backslash sum\_\{w\; \backslash in\; N\_G(v)\}\; \backslash mathcal\{M\}(G-v-w,\; x),$$

which is a well known recursive formula for the matching polynomial.

The following theorem, which is a generalization of (1.8), plays a crucial role in our proofs in Section 3.

Theorem 2.2. Let $G$ be a graph. For any subset $W$ of $V(G)$,

$$\mathcal{M}(S(G)-W,x)=x^{\mathrm{\mid }E(G)\mathrm{\mid }-\mathrm{\mid }V(G)\mathrm{\mid }+\mathrm{\mid }W\mathrm{\mid }}\mathfrak{M}(G-W,x^2{\mathbf{1}}_{G-W}-{\mathbf{d}}_{G,G-W})\mathrm{.}\backslash mathcal\{M\}(S(G)\; -\; W,\; x)\; =\; x^\{|E(G)|\; -\; |V(G)|\; +\; |W|\}\; \backslash mathfrak\{M\}(G\; -\; W,\; x^2\; \backslash mathbf\{1\}\_\{G-W\}\; -\; \backslash mathbf\{d\}\_\{G,G-W\}).$$

Proof. For simplicity, let $k = |V(G) \setminus W|$ and $m = |E(G)|$. We prove the assertion by induction on $k$. If $V(G) \setminus W = {u}$ for some vertex $u \in V(G)$, then $S(G) - W$ consists of a star on $d_G(u) + 1$ vertices and $|E(G)| - d_G(u)$ isolated vertices. Therefore,

$$\mathcal{M}(S(G)-W,x)=x^{m+1}-d_G(u)x^{m-1}\backslash mathcal\{M\}(S(G)\; -\; W,\; x)\; =\; x^\{m+1\}\; -\; d\_G(u)x^\{m-1\}$$

and

$$\mathfrak{M}(G-W,x^2\mathbf{1}-\mathbf{d})=x^2-d_G(u)\mathrm{.}\backslash mathfrak\{M\}(G\; -\; W,\; x^2\; \backslash mathbf\{1\}\; -\; \backslash mathbf\{d\})\; =\; x^2\; -\; d\_G(u).$$ ---PAGE_BREAK---

So, the claimed equality holds for $k=1$. Assume that $k \ge 2$. Choose a vertex $u \in V(G) \setminus W$ and let $H = S(G) - W - u$. By Lemma 2.1, the induction hypothesis and (2.4), we have

Hence, in order to complete the induction step, it suffices to prove that

$$(2.5)d_G(u)x\mathcal{M}(H,x)=x^2\sum _{v\in N_{S(G)-W}(u)}\mathcal{M}(H-v,x)-\sum _{v\in N_{G-W}(u)}\mathcal{M}(H-v,x)\mathrm{.}(2.5)\; \backslash qquad\; d\_G(u)x\backslash mathcal\{M\}(H,x)\; =\; x^2\; \backslash sum\_\{v\; \backslash in\; N\_\{S(G)-W\}(u)\}\; \backslash mathcal\{M\}(H-v,x)\; -\; \backslash sum\_\{v\; \backslash in\; N\_\{G-W\}(u)\}\; \backslash mathcal\{M\}(H-v,x).$$

To establish (2.5), let $N_G(u) \cap W = {a_1, \dots, a_s}$ and $N_G(u) \setminus W = {b_1, \dots, b_t}$. Also, for $i=1, \dots, s$, let $a'i$ be the vertex of $S(G)$ corresponding to the edge ${u, a_i}$ of $G$ and, for $j=1, \dots, t$, let $b'j$ be the vertex of $S(G)$ corresponding to the edge ${u, b_j}$ of $G$. Notice that, if one of $N_G(u) \cap W$ and $N_G(u) \setminus W$ is empty, then we may derive (2.5) by the same discussion as below. We have $d_G(u) = s+t$ and $N{S(G)-W}(u) = N{S(G)}(u) = {a'_1, \dots, a'_s, b'_1, \dots, b'_t}$. The structure of $H$ is illustrated in Figure 1.

Figure 1. The structure of $H$.

We have $d_H(a'_i) = 0$ for $i = 1, \dots, s$ and $d_H(b'_j) = 1$ for $j = 1, \dots, t$. By applying (2.4) for $a'_i$ and $b'_j$, we find that

$$\mathcal{M}(H,x)=x\mathcal{M}(H-a_i^{\mathrm{\prime }},x)\backslash mathcal\{M\}(H,x)\; =\; x\backslash mathcal\{M\}(H-a\text{'}\_i,x)$$ ---PAGE_BREAK---

and

Therefore,

which is exactly (2.5). This completes the proof. □

In what follows, we prove some results about the Laplacian matching polynomial by analysing the structures of the subdivision of graphs. The following consequence immediately follows from Theorem 2.2 and the first statement in (1.1). It worth to mention that the following result is proved in [17] for a different expression of the Laplacian matching polynomial.

Corollary 2.3. Let $G$ be a graph. Then

$$\mathcal{M}(S(G),x)=x^{\mathrm{\mid }E(G)\mathrm{\mid }-\mathrm{\mid }V(G)\mathrm{\mid }}\mathcal{L}\mathcal{M}(G,x^2)\mathrm{.}\backslash mathcal\{M\}(S(G),\; x)\; =\; x^\{|E(G)|-|V(G)|\}\; \backslash mathcal\{L\}\backslash mathcal\{M\}(G,\; x^2).$$

In particular, the zeros of $\mathcal{L}\mathcal{M}(G, x)$ are nonnegative real numbers.

For a graph $G$, it is proved that $\mathcal{L}\mathcal{M}(G, x) = \varphi(L(G), x)$ if and only if $G$ is a forest [11]. Since $0$ is an eigenvalue of $L(G)$, we deduce that $\mathcal{L}\mathcal{M}(G, 0) = 0$ if $G$ is a forest. From (1.4), we get the combinatorial identity

$$\sum _{M\in \mathcal{M}(F)}(-1{)}^{\mathrm{\mid }M\mathrm{\mid }}(\prod _{v\in V(F)\setminus V(M)}{d}_F(v))=0\backslash sum\_\{M\; \backslash in\; \backslash mathcal\{M\}(F)\}\; (-1)^\{|M|\}\; \backslash left(\; \backslash prod\_\{v\; \backslash in\; V(F)\; \backslash setminus\; V(M)\}\; d\_F(v)\; \backslash right)\; =\; 0$$

for any forest F. The following theorem, which is proved in [17], gives a necessary and sufficient condition for 0 to be a root of the Laplacian matching polynomial. We present here a different proof for it.

Theorem 2.4 (Chen, Zhang [17]). Let $G$ be a connected graph. Then, $0$ is a root of $\mathcal{LM}(G, x)$ if and only if $G$ is a tree.

Proof. If $G$ is a tree, then $|E(G)| = |V(G)| - 1$ and so $\mathcal{LM}(G, x^2) = x\mathcal{M}(S(G), x)$ by Corollary 2.3, implying that $0$ is a root of $\mathcal{LM}(G, x)$. We prove that $0$ is not a root of $\mathcal{LM}(G, x)$ if $G$ is not a tree. For this, assume that $|E(G)| \ge |V(G)|$. One may easily consider $S(G)$ as a bipartite graph with the bipartition ${V(G), E(G)}$ after identifying each new vertex $v_e$ of $S(G)$ with its corresponding edge $e$ of $G$.

We claim that $S(G)$ has a matching that saturates the part $V(G)$. If $G$ contains a vertex $u$ with degree 1 and $e$ is the edge incident to $u$ in $G$, then it suffices to prove that $S(G-u)$ has a matching that saturates the part $V(G-u)$, since the union of such matching and the edge ${u, v_e}$ forms a matching of $S(G)$ that saturates the part $V(G)$. Thus, we may assume that $d_G(v) \ge 2$ for all vertices $v \in V(G)$. We are going to establish that $S(G)$ satisfies Hall's condition [2, Theorem 16.4]. For a subset $W$ of $V(G)$, we shall use $N_G(W)$ to denote the set of vertices of $G$ each of which is adjacent to a vertex in $W$ and $\partial_G(W)$ to denote the set of edges of $G$ each of which has exactly one endpoint in $W$. For any subset $U$ of the part $V(G)$, since $d_G(v) \ge 2$ for all vertices $v \in V(G),

$$(2.6)\mathrm{\mid }{\mathrm{\partial }}_{S(G)}(U)\mathrm{\mid }\ge 2\mathrm{\mid }U\mathrm{\mid }\mathrm{.}(2.6)\; \backslash qquad\; |\backslash partial\_\{S(G)\}(U)|\; \backslash ge\; 2|U|.$$ ---PAGE_BREAK---

On the other hand, $d_{S(G)}(v_e) = 2$ for each $e \in E(G)$, so

$$(2.7)\mathrm{\mid }{\mathrm{\partial }}_{S(G)}(N_{S(G)}(U))\mathrm{\mid }=2\mathrm{\mid }{N}_{S(G)}(U)\mathrm{\mid }\mathrm{.}(2.7)\; \backslash qquad\; |\backslash partial\_\{S(G)\}(N\_\{S(G)\}(U))|\; =\; 2|N\_\{S(G)\}(U)|.$$

Clearly, $|\partial_{S(G)}(N_{S(G)}(U))| \ge |\partial_{S(G)}(U)|$ which implies that $|N_{S(G)}(U)| \ge |U|$ using (2.6) and (2.7). This means that $S(G)$ satisfies Hall's condition, as required.

We proved that $S(G)$ has a matching that saturates the part $V(G)$. This means that the smallest power of $x$ in $\mathcal{M}(S(G), x)$ is $|E(G)| - |V(G)|$ by (2.1) and (2.2). In view of Corollary 2.3, $\mathcal{M}(S(G), x) = x^{|E(G)|-|V(G)|}\mathcal{LM}(G, x^2)$ which shows that the constant term in $\mathcal{LM}(G, x)$ is nonzero. So, 0 is not a root of $\mathcal{LM}(G, x)$. This completes the proof. $\square$

In the next theorem, we give a lower bound on the number of distinct zeros of the Laplacian matching polynomial.

Theorem 2.5. Let $G$ be a connected graph and let $\ell(G)$ be the length of the longest path in $G$. Then the number of distinct positive roots of $\mathcal{LM}(G, x)$ is at least equal to $\ell(G)$. Also, if $\delta(G) \ge 2$, then $\mathcal{LM}(G, x)$ has at least $\ell(G) + 1$ distinct positive roots.

Proof. For convenience, let $\ell = \ell(G)$. Denote by $\ell'$ the length of the longest path in $S(G)$. From (1.2), $\mathcal{M}(S(G), x)$ has at least $\ell' + 1$ distinct roots. By Corollary 2.3, $\mathcal{M}(S(G), x) = x^{|E(G)|-|V(G)|}\mathcal{LM}(G, x^2)$ which shows that $\mathcal{LM}(G, x^2)$ has at least $\ell'$ distinct nonzero roots. Since all roots of $\mathcal{LM}(G, x)$ are real and nonnegative by Corollary 2.3, it follows that $\mathcal{LM}(G, x)$ has at least $\lceil \ell'/2 \rceil$ distinct positive roots.

For each edge $e \in E(G)$, denote by $v_e$ the vertex of $S(G)$ corresponding to $e$. Let $w_0, w_1, \dots, w_\ell$ be a path in $G$. Then, $w_0, v_{e_1}, w_1, \dots, v_{e_\ell}, w_\ell$ is a path in $S(G)$ of length $2\ell$, where $e_i = {w_{i-1}, w_i} \in E(G)$ for $i=1, \dots, \ell$. Thus, $\ell' \ge 2\ell$ and so $\mathcal{LM}(G, x)$ has at least $\ell$ distinct positive roots.

Now, assume that $\delta(G) \ge 2$. This assumption allows us to consider a vertex $w' \in N_G(w_0) \setminus {w_1}$. Then, $S(G)$ contains the path $v_{e'}$, $w_0$, $v_{e_1}$, $w_1$, \dots, $v_{e_\ell}$, $w_\ell$ of length $2\ell + 1$, where $e' = {w', w_0} \in E(G)$. Therefore, $\ell' \ge 2\ell + 1$ and so $\mathcal{LM}(G, x)$ has at least $\lceil \ell'/2 \rceil \ge \ell + 1$ distinct positive roots. This completes the proof. $\square$

Remark 2.6. The second statement in Theorem 2.5 implies that, if $G$ is a graph with a Hamilton cycle, then the zeros of $\mathcal{LM}(G, x)$ are all distinct.

Given a graph $G$, it is known that $\mathcal{M}(G, x) = \varphi(A(G, x))$ if and only if $G$ is a forest [14]. Also, as we mentioned before, it is established that $\mathcal{LM}(G, x) = \varphi(L(G), x)$ if and only if $G$ is a forest [11]. Below, we present a general result which shows that the multivariate matching polynomial of a forest has a determinantal representation in terms of its adjacency matrix, which will be used in the next section.

Theorem 2.7. Let $F$ be a forest. Then $\mathfrak{M}(F, \mathbf{x}_F) = \det(\mathbf{X}_F - A(F))$, where $\mathbf{X}_F$ is a diagonal matrix whose rows and columns are indexed by $V(F)$ and the $(v,v)$-entry is $x_v$ for any vertex $v \in V(F)$. In particular, $\mathcal{M}(F, x) = \varphi(A(F), x)$ and $\mathcal{LM}(F, x) = \varphi(L(F), x)$.

Proof. We prove that $\mathfrak{M}(F, \mathbf{x}_F) = \det(\mathbf{X}_F - A(F))$ by induction on $|E(F)|$. The equality is trivially valid if $|E(F)| = 0$. So, assume that $|E(F)| \ge 1$. As $F$ is a forest, we may consider two vertices $u, v \in V(F)$ with $N_F(u) = {v}$. Without loss of generality, we may assume that the first row and column of $A(F)$ are corresponding to $u$ and the second row and column of $A(F)$ are corresponding to $v$. Expanding the determinant of $\mathbf{X}_F - A(F)$ along its first row, we obtain by the induction hypothesis and Lemma 2.1 that

as desired. The 'in particular' statement immediately follows from (2.2) and (2.3). $\square$ ---PAGE_BREAK---

Corollary 2.8. For a tree $T$, the multiplicity of $0$ as a root of $\mathcal{LM}(T, x)$ is $1$.

Proof. It is well known that the number of connected components of a graph $\Gamma$ is equal to the multiplicity of $0$ as a root of $\varphi(L(\Gamma), x)$ [3, Proposition 1.3.7]. So, the result follows from $\mathcal{LM}(T, x) = \varphi(L(T), x)$ which is given in Theorem 2.7. $\square$

3. THE LARGEST ZERO OF THE LAPLACIAN MATCHING POLYNOMIAL

The purpose of this section is to investigate the location of the largest zero of the Laplacian matching polynomial. We give a linear algebraic approach to study the largest zero of the Laplacian matching polynomial and present sharp upper and lower bounds on it. The assertions (1.5) and (1.7) are also proved in this section based on the linear algebraic approach.

Let $G$ be a connected graph and $u \in V(G)$. Let $T(G, u)$ be the path-tree of $G$ respect to the vertex $u$ which is introduced in Section 1. Consider two vectors $x_G = (x_v){v \in V(G)}$ and $x{T(G,u)} = (x_P){P \in V(T(G,u))}$ of indeterminates associated with $G$ and $T(G, u)$, respectively. For every vertex $P \in V(T(G, u))$, we may identify $x_P$ with $x{v(P)}$ in which $v(P)$ is the terminal vertex of the path $P$ in $G$. In such way, $G$ and $T(G, u)$ will be equipped with two vectors consisting of the same indeterminates, which are simply denoted by x when there is no ambiguity. In what follows, for every subgraph $H$ of $G$ and vertex $u \in V(H)$, we denote by $D_G(T(H, u))$ the diagonal matrix whose rows and columns are indexed by $V(T(H, u))$ and the $(P, P)$-entry is $d_G(v(P))$.

The univariate version of the following theorem, which is proved by Godsil [5], has a key role in the theory of the matching polynomial. Notice that, for a graph $G$ and a vertex $u \in V(G)$, $u$ is a path in $G$ and the corresponding vertex in $T(G, u)$ will also be referred to as $u$.

Theorem 3.1 (Amini [1]). Let $G$ be a connected graph and let $u \in V(G)$. Then

$$\frac{\mathfrak{M}(G-u,\mathit{x})}{\mathfrak{M}(G,\mathit{x})}=\frac{\mathfrak{M}(T(G,u)-u,\mathit{x})}{\mathfrak{M}(T(G,u),\mathit{x})},\backslash frac\{\backslash mathfrak\{M\}(G\; -\; u,\; \backslash boldsymbol\{x\})\}\{\backslash mathfrak\{M\}(G,\; \backslash boldsymbol\{x\})\}\; =\; \backslash frac\{\backslash mathfrak\{M\}(T(G,\; u)\; -\; u,\; \backslash boldsymbol\{x\})\}\{\backslash mathfrak\{M\}(T(G,\; u),\; \backslash boldsymbol\{x\})\},$$

and moreover, $\mathfrak{M}(G, \boldsymbol{x})$ divides $\mathfrak{M}(T(G, u), \boldsymbol{x})$.

For a connected graph $G$ and a vertex $u \in V(G)$, Theorem 3.1 and Theorem 2.7 yield that $\mathcal{M}(G, x)$ divides $\varphi(A(T(G, u)), x)$. Since all roots of the characteristic polynomial of a symmetric matrix are real, the first statement in (1.1) is obtained as an application of Theorem 3.1. For the Laplacian matching polynomial, we get the following result.

Corollary 3.2. Let $G$ be a connected graph, $H$ be a subgraph of $G$, and $u \in V(H)$. If $H$ is connected, then $\mathfrak{M}(H, x\mathbf{1}H - d{G,H})$ divides $\varphi(D_G(T(H, u)) + A(T(H, u)), x)$. In particular, $\varphi(D_G(T(G, u)) + A(T(G, u)), x)$ is divisible by $\mathcal{LM}(G, x)$ for every vertex $u \in V(G)$.

Proof. By Theorem 3.1, we find that $\mathfrak{M}(H, x\mathbf{1}H - d{G,H})$ divides $\mathfrak{M}(T(H, u), x\mathbf{1}H - d{G,H})$. It follows from Theorem 2.7 that

which establishes what we require. Since $\mathfrak{M}(G, x\mathbf{1}_G - d_G) = \mathcal{LM}(G, x)$ using (2.3), the ‘in particular’ statement immediately follows. $\square$

Remark 3.3. The matrix $D_G(T(G, u)) + A(T(G, u))$, which appeared in Corollary 3.2, is a symmetric diagonally dominant matrix with nonnegative diagonal entries, so all of its eigenvalues are nonnegative real numbers. Hence, Corollary 3.2 gives us another proof for the fact that all roots of the Laplacian matching polynomial are real and nonnegative which was also proved in Corollary 2.3. ---PAGE_BREAK---

It is well known that the largest zero of the matching polynomial of a graph is equal to the largest eigenvalue of the adjacency matrix of a path-tree of that graph. This fact is obtained by combining the Perron-Frobenius theorem [3, Theorem 2.2.1] and Theorems 2.7 and 3.1. The following theorem can be considered as an analogue of the fact. Indeed, the following theorem presents a linear algebra technique to treat with the largest zero of the Laplacian matching polynomial.

Theorem 3.4. Let $G$ be a connected graph, $H$ be a subgraph of $G$, and $u \in V(H)$. If $H$ is connected, then

$$(3.1)\lambda (\mathfrak{M}(H,x{\mathbf{1}}_H-{\mathbf{d}}_{G,H}))=\lambda (D_G(T(H,u))+A(T(H,u)))\mathrm{.}(3.1)\; \backslash qquad\; \backslash lambda(\backslash mathfrak\{M\}(H,\; x\backslash mathbf\{1\}\_H\; -\; \backslash mathbf\{d\}\_\{G,H\}))\; =\; \backslash lambda(D\_G(T(H,u))\; +\; A(T(H,u))).$$

In particular, $\lambda(\mathcal{LM}(G,x)) = \lambda(D_G(T(G,u)) + A(T(G,u)))$. Also, the largest root of $\mathcal{LM}(G,x)$ has the multiplicity 1.

Proof. We prove (3.1) by induction on $|V(H)|$. Clearly, (3.1) is valid for $|V(H)| = 1$. Assume that $|V(H)| \ge 2$. We first show that

To see (3.2), we apply Theorem 2.2 and (1.3) to get that

where $W = V(G) \setminus V(H)$. This clearly proves (3.2). Now, let $N_H(u) = {u_1, \dots, u_k}$ and let $H_i$ be the connected component of $H-u$ containing $u_i$ for $i=1, \dots, k$. By the induction hypothesis,

$$(3.3)\lambda (\mathfrak{M}(H_i,x{\mathbf{1}}_{H_i}-{\mathbf{d}}_{G,H_i}))=\lambda (D_G(T(H_i,u_i))+A(T(H_i,u_i)))(3.3)\; \backslash qquad\; \backslash lambda(\backslash mathfrak\{M\}(H\_i,\; x\backslash mathbf\{1\}\_\{H\_i\}\; -\; \backslash mathbf\{d\}\_\{G,H\_i\}))\; =\; \backslash lambda(D\_G(T(H\_i,\; u\_i))\; +\; A(T(H\_i,\; u\_i)))$$

for $i=1, \dots, k$. It is not hard to see the $k \times k$ block diagonal matrix whose $i$th block diagonal entry is $D_G(T(H_i, u_i)) + A(T(H_i, u_i))$, say $R$, is a principal submatrix of $D_G(T(H, u)) + A(T(H, u))$ with size $|T(H, u)|-1$. Hence, by the interlacing theorem [3, Corollary 2.5.2], it follows that $\lambda(R)$ is greater than or equal to the second largest eigenvalue of $D_G(T(H, u)) + A(T(H, u))$. Further, it follows from (3.3) and (3.2) that

Thus, $\lambda(\mathfrak{M}(H, x\mathbf{1}H - \mathbf{d}{G,H}))$ is strictly greater than the second largest eigenvalue of $D_G(T(H, u)) + A(T(H, u))$. On the other hand, Corollary 3.2 implies that $\lambda(\mathfrak{M}(H, x\mathbf{1}H - \mathbf{d}{G,H}))$ is a zero of $\varphi(D_G(T(H, u)) + A(T(H, u)), x)$. So, we conclude that $\lambda(\mathfrak{M}(H, x\mathbf{1}H - \mathbf{d}{G,H}))$ is the largest eigenvalue of $D_G(T(H, u)) + A(T(H, u))$. This completes the induction step and demonstrates that (3.1) holds.

For the 'in particular' statement, note that (3.1) and (2.3) yield that

$$\lambda (D_G(T(G,u))+A(T(G,u)))=\lambda (\mathfrak{M}(G,x{\mathbf{1}}_G-{\mathbf{d}}_G))=\lambda (\mathcal{L}\mathcal{M}(G,x)),\backslash lambda(D\_G(T(G,u))\; +\; A(T(G,u)))\; =\; \backslash lambda(\backslash mathfrak\{M\}(G,\; x\backslash mathbf\{1\}\_G\; -\; \backslash mathbf\{d\}\_G))\; =\; \backslash lambda(\backslash mathcal\{LM\}(G,x)),$$

and further, the connectedness of $G$ implies that $D_G(T(G,u)) + A(T(G,u))$ is an irreducible matrix with nonnegative entries, and consequently, its largest eigenvalue has the multiplicity 1 by the Perron-Frobenius theorem [3, Theorem 2.2.1]. $\square$ ---PAGE_BREAK---

Corollary 3.5. Let $G$ be a connected graph and $u \in V(G)$. Then

$$(3.4)\lambda (\mathcal{L}\mathcal{M}(G,x))\ge \lambda (L(T(G,u)))(3.4)\; \backslash qquad\; \backslash lambda(\backslash mathcal\{LM\}(G,x))\; \backslash ge\; \backslash lambda(L(T(G,u)))$$

with the equality holds if and only if $G$ is a tree.

Proof. We first recall the fact that a graph $\Gamma$ is bipartite if and only if $\varphi(L(\Gamma), x) = \varphi(Q(\Gamma), x)$ [3, Proposition 1.3.10]. For each $P \in V(T(G, u))$, we have $d_{T(G,u)}(P) \le d_G(v(P))$, where $v(P)$ is the terminal vertex of the path $P$ in $G$. Therefore, $R = D_G(T(G, u)) + A(T(G, u)) - Q(T(G, u))$ has nonnegative entries, and thus, Theorem 3.4, the Perron-Frobenius theorem [3, Theorem 2.2.1], and the above mentioned fact yield that

proving (3.4). If $G$ is a tree, then $G$ is isometric to $T(G, u)$ and since $\mathcal{LM}(G,x) = \varphi(L(G), x)$ by Theorem 2.7, the equality in (3.4) is attained. Conversely, assume that the equality in (3.4) holds. Consequently, the equality in (3.5) occurs, and hence, the Perron-Frobenius theorem [3, Theorem 2.2.1] implies that $R=0$. This means that $d_{T(G,u)}(P) = d_G(v(P))$ for each $P \in V(T(G, u))$. We assert that $G$ is a tree. Towards a contradiction, suppose that there is a cycle $C$ in $G$. As $G$ is connected, there is a path $P_1$ in $G$ which start at $u$, none of its internal vertices is on $C$, and $v(P_1) \in V(C)$. Fix $w \in N_G(v(P_1)) \cap V(C)$ and let $P_2$ be the path on $C$ between $v(P_1)$ and $w$ whose length is more that 1. If $P$ is the path between $u$ and $w$ formed by $P_1$ and $P_2$, then it is clear that $d_{T(G,u)}(P) < d_G(v(P))$. This contradiction completes the proof. $\square$

In the following consequence, we give some lower bounds on the largest zero of the Laplacian matching polynomial.

Corollary 3.6. Let $G$ be a connected graph. Then

$$\lambda (\mathcal{L}\mathcal{M}(G,x))\ge \max \{\mathrm{\Delta }(G)+1,\delta (G)+\sqrt{\mathrm{\Delta }(G)}\}\backslash lambda(\backslash mathcal\{LM\}(G,x))\; \backslash ge\; \backslash max\; \backslash left\backslash \{\; \backslash Delta(G)\; +\; 1,\; \backslash delta(G)\; +\; \backslash sqrt\{\backslash Delta(G)\}\; \backslash right\backslash \}$$

with the equality holds if and only if $G$ is a star.

Proof. Let $u \in V(G)$ be of degree $\Delta(G)$. Indeed, $d_{T(G,u)}(u) = d_G(u)$ and therefore $\Delta(T(G,u)) = \Delta(G)$. For each connected graph $\Gamma$, Proposition 3.9.3 of [3] states that $\lambda(L(\Gamma)) \ge \Delta(\Gamma) + 1$ with the equality holds if and only if $\Delta(\Gamma) = |V(\Gamma)| - 1$. By this fact and Corollary 3.5, we obtain that $\lambda(\mathcal{LM}(G,x)) \ge \lambda(L(T(G,u))) \ge \Delta(T(G,u)) + 1 = \Delta(G) + 1$, and moreover, the equality $\lambda(\mathcal{LM}(G,x)) = \Delta(G) + 1$ holds if and only if $G$ is a star.

For each connected graph $\Gamma$, the Perron-Frobenius theorem [3, Theorem 2.2.1] implies that $\lambda(A(\Gamma)) \ge \sqrt{\Delta(\Gamma)}$ with the equality holds if and only if $\Gamma$ is a star. Using this fact, Theorem 3.4, and the Weyl inequality [3, Theorem 2.8.1], we derive

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Suppose that the equality $\lambda(\mathcal{LM}(G, x)) = \delta(G) + \sqrt{\Delta(G)}$ holds. So, the equality in (3.6) is attained, and thus, $T(G, u)$ is a star. This implies that $G$ is a star, and then, $\lambda(\mathcal{LM}(G, x)) = \delta(G) + \sqrt{\Delta(G)}$ forces that $|V(G)| \le 2$. Since the equality $\lambda(\mathcal{LM}(G, x)) = \delta(G) + \sqrt{\Delta(G)}$ is valid for the stars $G$ on at most 2 vertices, the proof is complete. $\square$

In the following theorem, we establish (1.5) which slightly improves the second statement of Theorem 2.6 of [11].

Theorem 3.7. Let $G$ be a connected graph with $\Delta(G) \ge 2$ and let $\ell(G)$ be the length of the longest path in $G$. Then,

$$(3.7)\lambda (\mathcal{L}\mathcal{M}(G,x))\le \mathrm{\Delta }(G)+2\sqrt{\mathrm{\Delta }(G)-1}\cos \frac{\pi }{2\mathrm{\ell }(G)+2}(3.7)\; \backslash qquad\; \backslash lambda(\backslash mathcal\{LM\}(G,\; x))\; \backslash le\; \backslash Delta(G)\; +\; 2\backslash sqrt\{\backslash Delta(G)-1\}\; \backslash cos\; \backslash frac\{\backslash pi\}\{2\backslash ell(G)+2\}$$

with the equality holds if and only if $G$ is a cycle.

Proof. For simplicity, let $\Delta = \Delta(G)$ and $\ell = \ell(G)$. For every positive integers $d$ and $k \ge 2$, the Bethe tree $B_{d,k}$ is a rooted tree with $k$ levels in which the root vertex is of degree $d$, the vertices on levels $2, \dots, k-1$ are of degree $d+1$, and the vertices on level $k$ are of degree 1. By Theorem 7 of [13],

$$(3.8)\lambda (A(B_{d,k}))=2\sqrt{d}\cos \frac{\pi }{k+1}\mathrm{.}(3.8)\; \backslash qquad\; \backslash lambda(A(B\_\{d,k\}))\; =\; 2\backslash sqrt\{d\}\; \backslash cos\; \backslash frac\{\backslash pi\}\{k+1\}.$$

Let $u \in V(G)$. It is not hard to check that $T(G, u)$ is isomorphic to a subgraph of $B_{\Delta-1,2\ell+1}$. For this, it is enough to correspond $u \in V(T(G, u))$ to an arbitrary vertex on level $\ell+1$ in $B_{\Delta-1,2\ell+1}$. By applying Theorem 3.4, the Weyl inequality [3, Theorem 2.8.1], the interlacing theorem [3, Corollary 2.5.2], and (3.8), we derive

proving (3.7). Now, assume that the equality in (3.7) is achieved. Therefore, the equality in (3.9) occurs, and thus, the Perron-Frobenius theorem [3, Theorem 2.2.1] implies that $T(G, u)$ is isomorphic to $B_{\Delta-1,2\ell+1}$. Since $\Delta \ge 2$, one can easily obtain that $G$ is a cycle. Conversely, if $G$ is a cycle, then $T(G, u)$ is a path on $2\ell+1$ vertices. By Theorem 3.4 and (3.8), we get

$$\lambda (\mathcal{L}\mathcal{M}(G,x))=\lambda (D_G(T(G,u))+A(T(G,u)))=2+\lambda (A(B_{1,2\mathrm{\ell }+1}))=2+2\cos \frac{\pi }{2\mathrm{\ell }+2}\mathrm{.}\backslash lambda(\backslash mathcal\{LM\}(G,x))\; =\; \backslash lambda(D\_G(T(G,u))\; +\; A(T(G,u)))\; =\; 2\; +\; \backslash lambda(A(B\_\{1,2\backslash ell+1\}))\; =\; 2\; +\; 2\backslash cos\; \backslash frac\{\backslash pi\}\{2\backslash ell+2\}.$$

This completes the proof. $\square$

Stevanović [15] proved that the eigenvalues of the adjacency matrix of a tree $T$ are less than $2\sqrt{\Delta(T)-1}$. The corollary below gives an improvement of this upper bound for the subdivision of trees.

Corollary 3.8. Let $G$ be a graph with $\Delta(G) \ge 2$. Then

In particular, if $F$ is a forest with $\Delta(F) \ge 2$, then $\lambda(A(S(F))) < 1 + \sqrt{\Delta(F)-1}$. ---PAGE_BREAK---

Proof. It follows from Theorem 3.7 that $\lambda(\mathcal{LM}(G, x)) < \Delta(G) + 2\sqrt{\Delta(G)-1}$. Moreover, it follows from Corollary 2.3 that $\lambda(\mathcal{M}(S(G), x)) = \sqrt{\lambda(\mathcal{LM}(G, x))}$. From these, we find that

proving (3.10). As the subdivision of a forest is a forest, the 'in particular' statement follows from Theorem 2.7 and (3.10). $\square$

Remark 3.9. Note that $\Delta(S(G)) = \Delta(G)$ for every graph $G$ with $\Delta(G) \ge 2$. So, for the subdivision of a graph with the maximum degree at least 2, the upper bound which appears in (3.10) is sharper than the upper bound that comes from (1.1).

We demonstrated in Theorem 3.4 that the largest zero of the Laplacian matching polynomial has the multiplicity 1. In the following theorem, we prove the remaining statements of (1.7) as analogues of the results given in (1.3).

Theorem 3.10. Let $G$ be a graph and let $n = |V(G)|$. For each edge $e \in E(G)$, the zeros of $\mathcal{LM}(G, x)$ and $\mathcal{LM}(G-e, x)$ interlace in the sense that, if $\alpha_1 \le \dots \le \alpha_n$ and $\beta_1 \le \dots \le \beta_n$ are respectively the zeros of $\mathcal{LM}(G, x)$ and $\mathcal{LM}(G-e, x)$, then $\beta_1 \le \alpha_1 \le \beta_2 \le \alpha_2 \le \dots \le \beta_n \le \alpha_n$. Also, if $G$ is connected, then $\lambda(\mathcal{LM}(G, x)) > \lambda(\mathcal{LM}(H, x))$ for any proper subgraph $H$ of $G$.

Proof. Fix an edge $e \in E(G)$ and denote by $v_e$ the vertex of $S(G)$ corresponding to $e$. Let $\alpha_1 \le \dots \le \alpha_n$ and $\beta_1 \le \dots \le \beta_n$ be the zeros of $\mathcal{LM}(G, x)$ and $\mathcal{LM}(G-e, x)$, respectively. Corollary 2.3 yields that $\sqrt{\alpha_1} \le \dots \le \sqrt{\alpha_n}$ is the end part of the nondescending sequence which consists of all the zeros of $\mathcal{M}(S(G), x)$ and $\sqrt{\beta_1} \le \dots \le \sqrt{\beta_n}$ is the end part of the nondescending sequence which consists of all the zeros of $\mathcal{M}(S(G-e), x)$. As $S(G-e) = S(G) - v_e$, it follows from (1.3) that the zeros of $\mathcal{M}(S(G), x)$ and $\mathcal{M}(S(G-e), x)$ interlace. So, we find that

$$\sqrt{{\beta }_1}\le \sqrt{{\alpha }_1}\le \sqrt{{\beta }_2}\le \sqrt{{\alpha }_2}\le \cdots \le \sqrt{{\beta }_n}\le \sqrt{{\alpha }_n}\backslash sqrt\{\backslash beta\_1\}\; \backslash le\; \backslash sqrt\{\backslash alpha\_1\}\; \backslash le\; \backslash sqrt\{\backslash beta\_2\}\; \backslash le\; \backslash sqrt\{\backslash alpha\_2\}\; \backslash le\; \backslash dots\; \backslash le\; \backslash sqrt\{\backslash beta\_n\}\; \backslash le\; \backslash sqrt\{\backslash alpha\_n\}$$

which means that $\beta_1 \le \alpha_1 \le \beta_2 \le \alpha_2 \le \dots \le \beta_n \le \alpha_n$, as desired.

Now, assume that $G$ is connected. Let $H$ be a proper subgraph of $G$ and let $u \in V(H)$. As $T(H, u)$ is a proper subgraph of $T(G, u)$, if $R$ denotes the submatrix of $D_G(T(G, u)) + A(T(G, u))$ corresponding to the vertices in $V(T(H, u))$, then $R - (D_H(T(H, u)) + A(T(H, u)))$ is a nonzero matrix with nonnegative entries. So, by applying Theorem 3.4 and the Perron-Frobenius theorem [3, Theorem 2.2.1], we get

Remark 3.11. For every graph $G$ and real number $\alpha$, let $m_G(\alpha)$ denote the multiplicity of $\alpha$ as a root of $\mathcal{LM}(G,x)$. As a consequence of Theorem 3.10, we have $|m_G(\alpha) - m_{G-e}(\alpha)| \le 1$ for each edge $e \in E(G)$.

It is known that among all trees with a fixed number of vertices the path has the smallest value of the largest Laplacian eigenvalue [12]. The following result can be considered as an analogue of this fact and is obtained from Theorems 2.7 and 3.10.

Corollary 3.12. Let $P_n$ and $K_n$ be the path and complete graph on $n$ vertices, respectively. For any connected graph $G$ on $n$ vertices which is not $P_n$ and $K_n$,

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4. CONCLUDING REMARKS

In this paper, we have discovered some properties of the location of zeros of the Laplacian matching polynomial. Most of our results can be considered as analogues of known results on the matching polynomial. Comparing to the matching polynomial, the Laplacian matching polynomial contains not only the information of the sizes of matchings in the graph but also the vertex degrees of the graph. Hence, it seems to be that more structural properties of graphs can be reflected by the Laplacian matching polynomial rather than the matching polynomial. For an instance, 0 is a root of $LM(G,x)$ if and only if $G$ is a forest, in while 0 is a root of $M(G,x)$ if and only if $G$ has no perfect matchings.

More interesting facts about the Laplacian matching polynomial can be concerned in further. For example, one may focus on the multiplicities of zeros of the Laplacian matching polynomial as there are many results on the multiplicities of zeros of the matching polynomial. In view of Remark 3.11, for every graph $G$ and real number $\alpha$, one may divide $E(G)$ into three subsets based on how the multiplicity of $\alpha$ changes when an edge of $G$ is removed. The corresponding problem about the matching polynomial is investigated by Chen and Ku [8]. Also, it is a known result that the multiplicity of a zero of the matching polynomial is at most the path partition number of the graph, that is, the minimum number of vertex disjoint paths required to cover all the vertices of the graph [4, Theorem 6.4.5]. It seems to be an interesting problem to find a sharp upper bound on the multiplicity of a zero of the Laplacian matching polynomial.

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