Spanning trees and even integer eigenvalues of graphs

TL;DR

This paper investigates the relationship between spanning trees and the eigenvalues of Laplacian and signless Laplacian matrices of graphs, establishing conditions under which certain eigenvalues are absent or have limited multiplicity, extending previous graph spectral results.

Contribution

It provides new conditions linking the divisibility of the number of spanning trees to the eigenvalues of graph Laplacians, extending prior spectral graph theory results.

Findings

If $ au(G)$ is not divisible by 4, then $L(G)$ has no even integer eigenvalues.

Under the same condition, $Q(G)$ has no integer eigenvalues $ eq 2 mod 4$.

The multiplicity of even integer eigenvalues of $Q(G)$ is bounded by the 2-adic valuation of $ au(G)$.

Abstract

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\tau(G)$ is not divisible by $4$, then (i) $L(G)$ has no even integer eigenvalue, (ii) $Q(G)$ has no integer eigenvalue $\lambda\equiv2\pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $\lambda\equiv0\pmod4$ and such an eigenvalue is simple. As a consequence, we extend previous results by Gutman and Sciriha and by Bapat on the nullity of adjacency matrices of the line graphs. We also show that if $\tau(G)=2^ts$ with $s$ odd, then the multiplicity of any even integer eigenvalue of $Q(G)$ is at most $t+1$. Among other things, we prove that if $L(G)$ or $Q(G)$ has an even integer eigenvalue of multiplicity at least $2$, then $\tau(G)$ is divisible by $4$. As a very special case of this result, a conjecture by Zhou et al. [On the nullity of connected graphs with least eigenvalue at least $-2$, Appl. Anal. Discrete Math. 7 (2013), 250--261] on the nullity of adjacency matrices of the line graphs of unicyclic graphs follows.