A graph for which the second largest distance eigenvalue is less than (-3+√(5))/(2) is chordal

2023/07/03 by Guo, Haiyan, Zhou, Bo
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.00917

Abstract

Let G be a connected graph with vertex set V(G). The distance, d_G(u,v), between vertices u and v in G is defined as the length of a shortest path between u and v in G. The distance matrix of G is the matrix D(G)=(d_G(u,v))_{u,v∈ V(G)}. The second largest distance eigenvalue of G is the second largest one in the spectrum of D(G). We show that any connected graph with the second largest distance eigenvalue less than (-3+√(5))/(2) is chordal, and characterize those bicyclic graphs and split graphs with the second largest distance eigenvalue less than -(1)/(2).

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