Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues

TL;DR

The paper studies the extremal behavior of the signless Laplacian spectrum by considering $S_2(G)=q_1(G)+q_2(G)$, the sum of the two largest signless Laplacian eigenvalues, and the derived objective $f(G)=e(G)+3-S_2(G)$. Leveraging the additive compound matrix $ abla_2(A)$, which connects sums of eigenvalues to spectral sums via $C_2(I_n+tA)$, the authors compare $f(G)$ across graph classes using interlacing and subgraph-decomposition techniques. The main result is that for any graph with $e(G)$ edges, the unique minimizer of $f(G)$ is the graph $K^+_{1,e(G)-1}$, extending prior vertex-based extremal results to the edge-count setting. This sharpens the understanding of extremal structures in the signless Laplacian spectrum and provides a precise edge-driven benchmark for $f(G)$.

Abstract

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Very recently, Zhou, He and Shan proved that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ among graphs on $n$ vertices. In this paper, we prove that $K^+_{1,e(G)-1}$ is the unique graph with minimum value of $f(G)$ among graphs with $e(G)$ edges.

Figures (3)

• Figure 1: Extremal graphs
• Figure 2: Extremal graphs with $c(G)=0$ and $1$
• Figure 3: All the trees with $e(G)=4$.

Theorems & Definitions (20)

• Theorem 1.1: LG
• Theorem 1.2
• Theorem 1.3: ZHS
• Theorem 1.4
• Conjecture 1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.1: MF
• Lemma 2.2: ZHS