Constructing Cospectral Vertices Through Orbits of Subgraphs

TL;DR

The paper develops a general, construction-based method to produce cospectral vertices in undirected graphs by combining two copies of a graph $G$ with a third graph $H$ and connecting them to respect automorphism orbits of $G$; the resulting pair of fixed vertices $v_c^1$ and $v_c^2$ are $A$-cospectral in the assembled graph $\widetilde{G}$. An analogous construction yields $L$-cospectral vertices, and a cospectrality-preserving modification extends the framework to generate new graphs with the same cospectral pair. The authors provide a clear sufficient condition for strong cospectrality (simple eigenvalues induced by $G$) and address potential degeneracies with a leaf-attachment lemma to reduce eigenvalue multiplicities. These results give a flexible toolkit for generating cospectral and strongly cospectral vertices and connect to latent symmetry concepts and Laplacian generalizations, with implications for spectral graph theory and quantum walk applications.

Abstract

A constructive method is given for obtaining cospectral vertices in undirected graphs, along with an operation that preserves this construction. We prove that the construction yields cospectral vertices, as well as strongly cospectral vertices under additional conditions. Furthermore, we generalize cospectral vertices to the case of the graph Laplacian and provide an analogous construction.

Figures (7)

• Figure 1: A graph containing non-symmetric cospectral vertices $v_1$ and $v_2$ colored in blue.
• Figure 2: Construction \ref{['cons:A_cospectral_constructive']} illustrated. The graphs $G^1$ and $G^2$ are divided into orbits under the action of $\mathrm{Aut}(G, v_c)$, shown by shades of gray, where the darkest gray contains only $v_{c}$, the fixed vertex of the construction. The vertices $v_{1}$, $v_{4}$ and $v_{2}$, $v_{3}$ are in the same orbit. For every edge from a vertex in $G^1$ to a vertex $v'_\alpha \in V(H)$, there is a corresponding edge to $v'_\alpha \in V(H)$ from a vertex of $G^2$ in the same orbit. Theorem \ref{['thm:A_cospectral_constructive']} proves that the vertices $v_{c}^1$ and $v_{c}^2$ are $A$-cospectral in the union graph $\widetilde{G}$.
• Figure 3: The red subgraphs together with the blue vertices are isomorphic to one another and represent the subgraphs $G^1$ and $G^2$ of Construction \ref{['cons:A_cospectral_constructive']}. Their connections to the black vertices in the middle, which represent the subgraph $H$, satisfy the conditions of Construction \ref{['cons:A_cospectral_constructive']} and so by Theorem \ref{['thm:A_cospectral_constructive']}, vertices $v_1^1$ and $v_1^2$ are $A$-cospectral.
• Figure 4: The red subgraphs correspond to $G^1$ and $G^2$ in Construction \ref{['cons:A_cospectral_constructive']}, and the blue vertices correspond to the $A$-cospectral vertices.
• Figure 5: Construction \ref{['cons:modify_cospectral_construction']} applied to a graph obtained from Construction \ref{['cons:A_cospectral_constructive']}. Here, the red subgraphs correspond to $G^1$ and $G^2$ in Construction \ref{['cons:A_cospectral_constructive']}, with the $A$-cospectral vertices indicated in blue. The green edges are those that are added in applying Construction \ref{['cons:modify_cospectral_construction']} in two different ways. It is easy to see that the two graphs are not isomorphic (in the right graph the two vertices with degrees 4 and 5 are connected, whereas in the left graph they are not). By Lemma \ref{['lem:modify_cospectral_construction']}, the blue vertices are still $A$-cospectral in both graphs.

Theorems & Definitions (28)

• Definition 2.1: Cospectral vertices
• Theorem 2.2
• Definition 2.3: Strongly cospectral vertices Strongly_Cospectral
• Definition 2.4: $A$-cospectral and $L$-cospectral vertices
• Remark 2.5
• Theorem 3.1
• proof
• Example 3.2
• Example 3.3
• Remark 3.4