Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the location of zeros of the Laplacian matching polynomials of graphs   The Laplacian matching polynomial of a graph $G$, denoted by
$\mathscr{L\hspace{-0.7mm}M}(G,x)$, 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 $\mathscr{L\hspace{-0.7mm}M}(G, x)$ if and only if
$G$ is a tree. We prove that the number of distinct positive zeros of
$\mathscr{L\hspace{-0.7mm}M}(G,x)$ is at least equal to the length of the
longest path in $G$. It is also established that the zeros of
$\mathscr{L\hspace{-0.7mm}M}(G,x)$ and $\mathscr{L\hspace{-0.7mm}M}(G-e,x)$
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
$\mathscr{L\hspace{-0.7mm}M}(G,x)$ and particularly to give tight upper and
lower bounds on it.

18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas