Spectral Radius and Degree Sequence of a Graph

TL;DR

This paper derives sharp upper bounds for the spectral radius of a connected graph based on its degree sequence, generalizing previous results and providing insights into the relationship between degree sequences and spectral properties.

Contribution

It introduces a new, generalized upper bound for the spectral radius of a graph using its degree sequence, extending prior bounds and enhancing spectral graph theory understanding.

Findings

Provides a sharp upper bound for spectral radius based on degree sequence.

Generalizes previous bounds for spectral radius.

Enhances understanding of spectral properties in relation to degree sequences.

Abstract

Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give a sharp upper bound for rho(G) by a function of d_1, d_2, ..., d_L, which generalizes a series of previous results.