Nordhaus-Gaddum Type Inequalities for Laplacian and Signless Laplacian Eigenvalues

Abstract

Let G be a graph with n vertices. We denote the largest signless Laplacian eigenvalue of G by q1(G) and Laplacian eigenvalues of G by µ1(G) ≥ · · · ≥ µn−1(G) ≥ µn(G) = 0. It is a conjecture on Laplacian spread of graphs that µ1(G) − µn−1(G) ≤ n − 1 or equivalently µ1(G) + µ1(G) ≤ 2n − 1. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph G, µ1(G)µ1(G) ≤ n(n − 1). Aouchiche and Hansen [A survey of Nordhaus-Gaddum type relations, Discrete Appl. Math. 161 (2013), 466-546] conjectured that q1(G) + q1(G) ≤ 3n − 4 and q1(G)q1(G) ≤ 2n(n − 2). We prove the former and disprove the latter by constructing a family of graphs Hn where q1(Hn)q1(Hn) is about 2.15n 2 + O(n).

Signless Laplacian eigenvalue problems of Nordhaus–Gaddum type

“…With regard to the signless Laplacian eigenvalues, Ashraf and Tayfeh-Rezaie [3] showed that q 1 (G) + q 1 (G) ≤ 3n − 4, which confirmed a conjecture posed by Aouchiche and Hansen [2]. In this paper, we study the similar problem on q 2 (G) + q 2 (G) as follows.…”

“…Notice that the inequality in Conjecture 2 is equivalent to µ 1 (G)+µ 1 (G) ≤ 2n−1 or µ n−1 (G) + µ n−1 (G) ≥ 1. Ashraf and Tayfeh-Rezaie [3] confirmed Conjecture 2 for bipartite graphs, and Chen and Das [6] confirmed Conjecture 2 for graphs with d 1 (G) − d n (G) ≤ n − 3 + 2/n. Very recently, Einollahzadeh and Karkhaneei [8] completely confirmed Conjecture 2.…”

Algebraic connectivity of the second power of a graph

“…Remark The item $$(\mathrm{iv})$$ of Theorem 1 was first proposed as a conjecture in [15, 16] and then was studied in multiple studies [1–7, 9–14, 16] and was recently proved in [8]. …”

Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues

“…In 2014, Ashraf et al [4] confirmed it for bipartite graphs and characterized the case when equality holds. Finally in 2021, Einollahzadeh and Karkhaneei [9] completely confirmed it.…”

“…, n. There are many results about it, for more details see [1,6,19,20,22,23]. For the signless Laplacian eigenvalues, Ashraf and Tayfeh-Rezaie [4] showed that q 1 (G) + q 1 (G) 󰃑 3n − 4. Huang and Lin [17] proved that n − 2 󰃑 q 2 (G) + q 2 (G) 󰃑 2n − 4.…”