An Upper Bound on Generalized Cospectral Mates of Oriented Graphs Using Skew-Walk Matrices

TL;DR

The paper investigates how many non-isomorphic generalized cospectral mates oriented graphs can have under the generalized skew spectrum, introducing the skew-walk matrix $W(D)$ and the framework of rational orthogonal matrices in $\\Gamma(D)$. It proves a tight upper bound: for $D\\in \\mathcal{F}_n$, the number of non-isomorphic generalized cospectral mates is at most $2^k-1$, where $k$ is the number of distinct odd prime factors of $\\det W(D)$, with the bound realized in explicit examples. The analysis leverages Smith normal form, invariant factors, and modular rank arguments to show distinct levels yield distinct (non-isomorphic) mates, while identical levels force isomorphism. A corollary yields a WDGSS criterion when $2^{-\\lfloor n/2\\rfloor}\\det W(D)$ is an odd prime, and the work connects spectral graph characterization to controllability, suggesting broader extensions to undirected graphs and alternative dynamical settings.

Abstract

Let $D$ be an oriented graph with skew adjacency matrix $S(D)$. Two oriented graphs $D$ and $C$ are said to share the same generalized skew spectrum if $S(D)$ and $S(C)$ have the same eigenvalues, and $J-S(D)$ and $J-S(C)$ also have the same eigenvalues, where $J$ is the all-ones matrix. Such graphs that are not isomorphic are generalized cospectral mates. We derive tight upper bounds on the number of non-isomorphic generalized cospectral mates an oriented graph can admit, based on arithmetic criteria involving the determinant of its skew-walk matrix. As a special case, we also provide a criterion for an oriented graph to be weakly determined by its generalized skew spectrum (WDGSS), that is, its only generalized cospectral mate is its transpose. These criteria relate directly to the controllability of graphs, a fundamental concept in the control of networked systems, thereby connecting spectral characterization of graphs to graph controllability.

Figures (2)

• Figure 1: Two generalized cospectral oriented graphs.
• Figure 2: An example oriented graph that is WDGSS and satisfies the condition of Corollary \ref{['corollary:wdgss']}.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1
• proof