Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Enumeration of Generalized Cospectral Mates of Graphs

Muhammad Razaa, Obaid Ullah Ahmad, Mudassir Shabbir, Waseem Abbas

TL;DR

The paper tackles the problem of how many non-isomorphic graphs can share the generalized spectrum $(\Spec(G),\Spec(\overline{G}))$ by leveraging the Smith Normal Form of the walk matrix $W(G)$ and the level $\ell(Q)$ of regular rational orthogonal matrices. Focusing on the graph family $\mathcal{F}_n$ defined via SNF constraints, it proves that each possible level corresponds to at most one generalized cospectral mate, and provides an explicit upper bound of $\left(\prod_j k_j\right) - 1$ where $d_n(W(G))=\prod_j p_j^{k_j}$. The authors validate the bound with a tight $n=10$ example and report numerical evidence that a sizeable portion of random graphs lie in $\mathcal{F}_n$, suggesting broad applicability. They also discuss potential extensions to other matrices and to relaxing the defining constraints, outlining directions for future research in spectral graph theory. $W(G)$, $d_n(W(G))$, $\mathcal{F}_n$, and $\ell(Q)$ are central to the methodology and results.

Abstract

This paper investigates the enumeration of generalized cospectral mates of simple graphs, where the generalized spectrum consists of the spectra of a graph and its complement. Moving beyond the classical problem of identifying graphs determined by their generalized spectrum, we address the more quantitative question of how many non-isomorphic graphs can share the same generalized spectrum. Our approach is based on arithmetic constraints derived from the Smith Normal Form (SNF) of the walk matrix $W(G)$, which lead to a tight upper bound on the number of generalized cospectral mates of a graph. Our upper bound applies to a much broader class of graphs than those previously shown to have no generalized cospectral mates (determined by generalized spectrum). Consequently, this work extends the family of graphs for which strong and informative spectral uniqueness

On the Enumeration of Generalized Cospectral Mates of Graphs

TL;DR

The paper tackles the problem of how many non-isomorphic graphs can share the generalized spectrum by leveraging the Smith Normal Form of the walk matrix and the level of regular rational orthogonal matrices. Focusing on the graph family defined via SNF constraints, it proves that each possible level corresponds to at most one generalized cospectral mate, and provides an explicit upper bound of where . The authors validate the bound with a tight example and report numerical evidence that a sizeable portion of random graphs lie in , suggesting broad applicability. They also discuss potential extensions to other matrices and to relaxing the defining constraints, outlining directions for future research in spectral graph theory. , , , and are central to the methodology and results.

Abstract

This paper investigates the enumeration of generalized cospectral mates of simple graphs, where the generalized spectrum consists of the spectra of a graph and its complement. Moving beyond the classical problem of identifying graphs determined by their generalized spectrum, we address the more quantitative question of how many non-isomorphic graphs can share the same generalized spectrum. Our approach is based on arithmetic constraints derived from the Smith Normal Form (SNF) of the walk matrix , which lead to a tight upper bound on the number of generalized cospectral mates of a graph. Our upper bound applies to a much broader class of graphs than those previously shown to have no generalized cospectral mates (determined by generalized spectrum). Consequently, this work extends the family of graphs for which strong and informative spectral uniqueness
Paper Structure (6 sections, 18 theorems, 40 equations, 1 figure, 1 table)

This paper contains 6 sections, 18 theorems, 40 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Arithmetic Constraints on Matrices in $O_n(\mathbb{Q})$
  4. Proofs of Theorem \ref{['theorem:level-unique']} and Theorem \ref{['theorem:main']}
  5. An Example and Numerical Evaluation
  6. Conclusion

Key Result

Theorem 1

Let $G \in \mathcal{F}_n$, and let $H_1$ and $H_2$ be graphs that are generalized cospectral with $G$. Let $Q_1$ and $Q_2$ be the rational orthogonal matrices satisfying $Q_i^T A(G) Q_i = A(H_i)$ and $Q_i e = e$ for $i=1,2$. If $\ell(Q_1) = \ell(Q_2)$, then $H_1 \cong H_2$.

Figures (1)

  • Figure 1: Graph $G$ and all of its generalized cospectral mates $H_1, H_2$ and $H_3$.

Theorems & Definitions (30)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 20 more