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Spectral properties of $\mathcal{C}$-graphs

Santanu Mandal, Ranjit Mehatari

TL;DR

This work introduces \(\mathcal{C}\)-graphs, a broad subclass of cographs generated by an even-length sequence \((\alpha_1,\dots,\alpha_{2k})\) and analyzed via an equitable partition to study their adjacency spectra. The authors derive the inertia and the multiplicities of the trivial eigenvalues \(-1\) and \(0\), with explicit formulas: \(n_-(A)=\sum_{i=1}^k\alpha_{2i-1}\), \(n_0(A)=\sum_{i=1}^k\alpha_{2i}-k\), and \(n_+(A)=k\); they also show that \(-1\) is an eigenvalue of the quotient matrix iff \(\alpha_2=1\). A key result is an eigenvalue-free interval for nontrivial eigenvalues: none lie in $\left[\frac{-1-\sqrt{2}}{2}, \frac{-1+\sqrt{2}}{2}\alpha_{\min}\right]$, where $\alpha_{\min}=\min_i \alpha_i$, with extremal eigenvalues bounded accordingly. The characteristic polynomial is given in closed form via a quotient-polynomial and a tridiagonal determinant, and an explicit example is worked out to illustrate the method. The work also discusses the uniqueness of the creation-sequence representation for even length and raises open questions about extending these spectral characterizations to all cographs.

Abstract

Assumed to be undirected, simple, and connected are all of the graphs in this study, and adjacency matrix $A$ serves as the associated matrix. In this paper we show that it is possible to relate a creation sequence for a type of cographs (we call it $\mathcal{C}$-graphs). Those cographs can be defined by a finite sequence of natural numbers. Using that sequence we obtain the inertia of the cograph under consideration. An extended eigenvalue-free set from $(-1,0)$ to $\big{[}\frac{-1-\sqrt{2}}{2}, -1)\cup (-1, 0) \cup (0, \frac{-1+\sqrt{2}}{2}α_{min}\big{]}$, (where $α_{min}\geq1$ is the smallest integer of the creation sequence) is obtained for the cographs under consideration. Additionally, an exact formula is found for the characteristic polynomial.

Spectral properties of $\mathcal{C}$-graphs

TL;DR

This work introduces -graphs, a broad subclass of cographs generated by an even-length sequence \((\alpha_1,\dots,\alpha_{2k})\) and analyzed via an equitable partition to study their adjacency spectra. The authors derive the inertia and the multiplicities of the trivial eigenvalues and , with explicit formulas: \(n_-(A)=\sum_{i=1}^k\alpha_{2i-1}\), \(n_0(A)=\sum_{i=1}^k\alpha_{2i}-k\), and \(n_+(A)=k\); they also show that is an eigenvalue of the quotient matrix iff . A key result is an eigenvalue-free interval for nontrivial eigenvalues: none lie in , where , with extremal eigenvalues bounded accordingly. The characteristic polynomial is given in closed form via a quotient-polynomial and a tridiagonal determinant, and an explicit example is worked out to illustrate the method. The work also discusses the uniqueness of the creation-sequence representation for even length and raises open questions about extending these spectral characterizations to all cographs.

Abstract

Assumed to be undirected, simple, and connected are all of the graphs in this study, and adjacency matrix serves as the associated matrix. In this paper we show that it is possible to relate a creation sequence for a type of cographs (we call it -graphs). Those cographs can be defined by a finite sequence of natural numbers. Using that sequence we obtain the inertia of the cograph under consideration. An extended eigenvalue-free set from to , (where is the smallest integer of the creation sequence) is obtained for the cographs under consideration. Additionally, an exact formula is found for the characteristic polynomial.
Paper Structure (9 sections, 17 theorems, 44 equations, 1 figure)

This paper contains 9 sections, 17 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Eigenvalues of $\mathcal{C}$-graphs
  3. Inertia of a $\mathcal{C}$-graph
  4. Eigenvalue-free interval for $\mathcal{C}$-graphs
  5. Characteristic polynomial
  6. Conclusions
  7. Acknowledgements
  8. Statements and Declarations
  9. Data availability

Key Result

Theorem 1.1

Let $G$ be a $\mathcal{C}$-graph with n vertices. If $G\in\mathcal{C}$ is generated in even number of steps, then there exists an unique integer partition $\{\alpha_1,\alpha_2,\ldots,\alpha_{2k}\}$ of $n$ such that $G=C(\alpha_1,\alpha_2,\ldots,\alpha_{2k})$. In other wards, the representation of $G

Figures (1)

  • Figure 1: Construction of the $\mathcal{C}$-graph $C(3,2,2,3)$

Theorems & Definitions (25)

  • Theorem 1.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3: Weyl's inequality
  • Theorem 2.4: Sylvester's law of inertia
  • ...and 15 more