Exploring Graphs with Distinct $M$-Eigenvalues: Product Operation, Wronskian Vertices, and Controllability

TL;DR

This paper investigates the spectral properties of a new graph product related to $M$-eigenvalues, introduces the concept of $M$-Wronskian vertices, and provides conditions for graph controllability and construction of $M$-cospectral graphs.

Contribution

It introduces a new graph product, characterizes when it has distinct $M$-eigenvalues, and develops methods for constructing $M$-cospectral graphs and analyzing controllability.

Findings

Derived necessary and sufficient conditions for $G\circ_C H$ to have distinct $M$-eigenvalues.

Introduced the concept of $M$-Wronskian vertices for graph analysis.

Provided criteria for $G\circ H$ to be $M$-controllable.

Abstract

Let $\mathcal{G}^M$ denote the set of connected graphs with distinct $M$-eigenvalues. This paper explores the $M$-spectrum and eigenvectors of a new product $G\circ_C H$ of graphs $G$ and $H$. We present the necessary and sufficient condition for $G\circ_C H$ to have distinct $M$-eigenvalues. Specifically, for the rooted product $G\circ H$, we present a more concise and precise condition. A key concept, the $M$-Wronskian vertex, which plays a crucial role in determining graph properties related to separability and construction of specific graph families, is investigated. We propose a novel method for constructing infinite pairs of non-isomorphic $M$-cospectral graphs in $\mathcal{G}^M$ by leveraging the structural properties of the $M$-Wronskian vertex. Moreover, the necessary and sufficient condition for $G\circ H$ to be $M$-controllable is given.