The least distance eigenvalue of the complements of graphs of diameter greater than three

2023/02/24 by Chen, Xu, Zhu, Yinfen, Wang, Guoping
#05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.13761

Abstract

Suppose G is a connected simple graph with the vertex set V( G ) = \ v₁,v₂,⋯ ,v_n \ . Let d_G( v_i,v_j ) be the least distance between v_i and v_j in G. Then the distance matrix of G is D( G ) =( d_ij ) _{n× n}, where d_ij=d_G( v_i,v_j ) . Since D( G ) is a non-negative real symmetric matrix, its eigenvalues can be arranged as λ₁(G)≥ λ₂(G)≥ ⋯ ≥ λ_n(G), where eigenvalue λ_n(G) is called the least distance eigenvalue of G. In this paper we determine the unique graph whose least distance eigenvalue attains maximum among all complements of graphs of diameter greater than three.

Related

• The distance spectrum of the complements of graphs of diameter greater than three
• Tricyclic graphs for which the second largest distance eigenvalue less than -(1)/(2)
• Graphs with at most three distance eigenvalues different from -1 and -2
• Distance-regular graphs with a few q-distance eigenvalues
• The spectra of the complements of graphs with given connectivity