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arXiv · 2504.04389

Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues

Zi-Ming Zhou·Zhi-Bin Du·Chang-Xiang He
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Abstract

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Very recently, Zhou, He and Shan proved that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ among graphs on $n$ vertices. In this paper, we prove that $K^+_{1,e(G)-1}$ is the unique graph with minimum value of $f(G)$ among graphs with $e(G)$ edges.

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Zi-Ming Zhou, Zhi-Bin Du, Chang-Xiang He. 2025-04-06. Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues. https://arxiv.org/abs/2504.04389

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