Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2412.18759 |
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Inhaltsangabe:
- 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.