The spectral radius of graphs without paths and cycles of specified length

TL;DR

This paper investigates the maximum spectral radius of graphs that exclude certain paths and cycles, providing exact bounds and raising open problems in spectral graph theory.

Contribution

It determines the maximum spectral radius for graphs without specific paths and even cycles, advancing understanding of spectral properties under structural constraints.

Findings

Maximum spectral radius for graphs without given paths

Tight bounds for graphs without even cycles

Open problems in spectral graph theory

Abstract

Let G be a graph with n vertices and mu(G) be the largest eigenvalue of the adjacency matrix of G. We study how large mu(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of natural open problems.