Spectral study of alliances in graphs

TL;DR

This paper explores the spectral properties of alliances in graphs, establishing tight bounds on alliance numbers and their relationships with spectral graph parameters like algebraic connectivity and spectral radius.

Contribution

It provides new tight bounds on various alliance numbers and links these to spectral graph parameters, enhancing understanding of graph alliances through spectral analysis.

Findings

Derived tight bounds on alliance numbers.

Established relationships between alliance numbers and spectral parameters.

Enhanced understanding of alliances via spectral graph theory.

Abstract

In this paper we obtain several tight bounds on different types of alliance numbers of a graph: (global) defensive alliance number, global offensive alliance number and global dual alliance number. In particular, we investigate the relationship between the alliance numbers of a graph and its algebraic connectivity, its spectral radius, and its Laplacian spectral radius.