Open problem on σ-invariant

Abstract

Let G be a graph of order n with m edges. Also let µ1 ≥ µ2 ≥ . . . ≥ µn-1 ≥ µn = 0 be the Laplacian eigenvalues of graph G and let σ = σ(G) (1 ≤ σ ≤ n) be the largest positive integer such that µσ ≥ 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 [8], that is, the characterization of all graphs with σ = 1.