Abstract
The Laplacian matching polynomial of a graph G, denoted by \(\mathscr {LM}(G,x)\), is a new graph polynomial whose all zeros 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 zero of \(\mathscr {LM}(G, x)\) if and only if G is a tree. We prove that the number of distinct positive zeros of \(\mathscr {LM}(G,x)\) is at least equal to the length of the longest path in G. It is also established that the zeros of \(\mathscr {LM}(G,x)\) and \(\mathscr {LM}(G-e,x)\) 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 \(\mathscr {LM}(G,x)\) and particularly to give tight upper and lower bounds on it.
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Funding
The research of the second author is supported by the National Natural Science Foundation of China with grant numbers 12171002 and 11871073. The research of the third author is supported by the Natural Science Foundation of Anhui Province with grant identifier 2008085MA03 and by the National Natural Science Foundation of China with grant number 12171002.
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Wan, JC., Wang, Y. & Mohammadian, A. On the location of zeros of the Laplacian matching polynomials of graphs. J Algebr Comb 56, 755–771 (2022). https://doi.org/10.1007/s10801-022-01130-5
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DOI: https://doi.org/10.1007/s10801-022-01130-5