Exploring graphs with distinct M-eigenvalues: Product operation, Wronskian vertices, and controllability

Abstract

Let $G^M$ 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, $M$-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 $G^M$ by leveraging the structural properties of $M$-Wronskian vertex. Moreover, the necessary and sufficient condition for $G\circ H$ to be $M$-controllable is given.