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- W4297582166 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 $lambdaequiv2pmod4$, and (iii) $Q(G)$ has at most one eigenvalue $lambdaequiv0pmod4$ 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." @default.
- W4297582166 created "2022-09-30" @default.
- W4297582166 creator A5008547727 @default.
- W4297582166 date "2012-01-16" @default.
- W4297582166 modified "2023-10-14" @default.
- W4297582166 title "Spanning trees and even integer eigenvalues of graphs" @default.
- W4297582166 doi "https://doi.org/10.48550/arxiv.1201.3221" @default.
- W4297582166 hasPublicationYear "2012" @default.
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