Open problem on σ-invariant

2017/11/18 by Das, Kinkar Ch., Mojallal, Seyed Ahmad
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.06906

Abstract

Let G be a graph of order n with m edges. Also let μ₁≥ μ₂≥ ⋯≥ μ_n-1≥ μ_n=0 be the Laplacian eigenvalues of graph G and let σ=σ(G) (1≤ σ≤ n) be the largest positive integer such that μ_σ≥ (2m)/(n). In this paper, we prove that μ₂(G)≥ (2m)/(n) for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in \citeKMT, that is, the characterization of all graphs with σ=1.

Related

• Partial characterization of graphs having a single large Laplacian\n eigenvalue
• Notes on the sum of powers of the signless Laplacian eigenvalues of graphs
• Upper bound for the Laplacian eigenvalues of a graph
• A lower bound of the least signless Laplacian eigenvalue of a graph
• Eigenvalue bounds for the signless p-Laplacian