Let G be a regular graph with m edges, and let μ 1, μ 2 denote the two largest eigenvalues of
AG, the adjacency matrix of G. We show that, if G is not complete, then μ 1 2+ μ 2 2≤ 2 (ω …

Abstract Brualdi–Hoffman–Turán-type problem asks what is the maximum spectral radius λ 1
(G) of an H-free graph G on m edges? This problem gives a spectral perspective on the …

This paper delves into three research directions, leveraging the Lovász ϑ-function of a
graph. First, it focuses on the Shannon capacity of graphs, providing new results that …

A fundamental result in extremal graph theory attributes to Mantel's theorem, which states
that every graph on $ n $ vertices with more than $\lfloor n^ 2/4\rfloor $ edges contains a …

We investigate two conjectured spectral graph theoretic strengthenings of Turán's theorem.
Let μ 1≥…≥ μ n denote the eigenvalues of a graph G with n vertices, m edges and clique …

This paper delves into three research directions, leveraging the Lovász $\vartheta $-function
of a graph. First, it focuses on the Shannon capacity of graphs, providing new results that …

In this paper, we generalize the log-Sobolev inequalities to Rényi–Sobolev inequalities by
replacing the entropy with the two-parameter entropy, which is a generalized version of …

In 2002, Nikiforov proved that for an $ n $-vertex graph $ G $ with clique number $\omega $,
edge number $ m $ and spectral radius $\lambda (G) $, there holds $\lambda (G)\leq\sqrt {2 …

We give a complete characterisation of the cubic graphs with no eigenvalues in the open
interval $(-1, 1) $. There are two infinite families, one due to Guo and Mohar [Linear Algebra …

Bollob\'as and Nikiforov conjectured that for any graph $ G\neq K_n $ with $ m $
edges\[\lambda_1^ 2+\lambda_2^ 2\le\bigg (1-\frac {1}{\omega (G)}\bigg) 2m\] where …