When (signless) Laplacian coefficients meet matchings of subdivision

Abstract

Let $G$ be a graph, whose subdivision is denoted by $S\left(G\right)$. Let ${\varphi }_L(G,x)$ be the characteristic polynomial of the Laplacian matrix of $G$. In 1974, Kelmans and Chelnokov (1974) gave a graph theoretical interpretation for the coefficients of ${\varphi }_L(G,x)$, in terms of the spanning forests of $G$. In this paper, we present another graph theoretical interpretation of the Laplacian coefficients by using the matching numbers of $S\left(G\right)$, generalizing the cases of trees and unicyclic graphs, which were established by Zhou and Gutman (2008) and Chen and Yan (2021), respectively. Analogously, a graph theoretical interpretation of the signless Laplacian coefficients is also presented, whose previous graph theoretical interpretation is based on the so-called TU-subgraphs (the spanning subgraphs whose components are trees or odd-unicyclic graphs) due to Cvetković et al. (2007). Some formulas related to the number of spanning trees are also given.