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

2025/04/06 by Zhou, Zi-Ming, Du, Zhi-Bin, He, Chang-Xiang
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.04389

Abstract

For a graph G, let S₂(G) be the sum of the first two largest signless Laplacian eigenvalues of G, and f(G)=e(G)+3-S₂(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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