The main vertices of a star set and related graph parameters

TL;DR

This paper introduces new spectral graph parameters based on main vertices of eigenvalues, explores their properties, and develops optimization tools to aid in graph isomorphism research.

Contribution

It defines new invariants related to main vertices of eigenvalues and formulates their determination as a combinatorial optimization problem.

Findings

Introduced minimum and maximum number of main vertices as new invariants.

Developed a simplex-like approach for computing these parameters.

Provided examples where these parameters match star set sizes.

Abstract

A vertex $v \in V(G)$ is called $\lambda$-main if it belongs to a star set $X \subset V(G)$ of the eigenvalue $\lambda$ of a graph $G$ and this eigenvalue is main for the graph obtained from $G$ by deleting all the vertices in $X \setminus \{v\}$; otherwise, $v$ is $\lambda$-non-main. Some results concerning main and non-main vertices of an eigenvalue are deduced. For a main eigenvalue $\lambda$ of a graph $G$, we introduce the minimum and maximum number of $\lambda$-main vertices in some $\lambda$-star set of $G$ as new graph invariant parameters. The determination of these parameters is formulated as a combinatorial optimization problem based on a simplex-like approach. Using these and some related parameters we develop new spectral tools that can be used in the research of the isomorphism problem. Examples of graphs for which the maximum number of $\lambda$-main vertices coincides with the cardinality of a $\lambda$-star set are provided.