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The spectral extrema of graphs of odd size forbidding $H(4,3)$ beyond the book graph

Abdul Basit Wani, S. Pirzada, Amir Rehman

TL;DR

This paper addresses the spectral extremal problem for $H(4,3)$-free graphs with an odd number of edges, restricting to graphs that exclude the book graph $K_2 \vee \frac{m-1}{2}K_1$. It proves a sharp upper bound on the spectral radius $\lambda(G)$ for odd $m\ge 58$, identifying the unique extremal graph $K_1 \vee (K_{1,(m-3)/2} \cup 2K_1)$ and showing $\lambda(G)$ equals the largest root $\tilde{\lambda}(m)$ of $x^4 - m x^2 - (m-3)x + m - 3 = 0$; the proof combines equitable partitions, quotient matrices, and edge-relocation arguments. The work situates these findings within the broader spectral Turán program and discusses connections to recent odd/even size results and potential future directions. Overall, it provides a precise extremal graph characterization and a sharp spectral bound under a natural forbidden-subgraph constraint, advancing understanding of how forbidding $H(4,3)$ shapes spectral properties.

Abstract

A graph is said to be $H$-free if it does not contain a subgraph isomorphic to $H$. The fish graph, denoted by $H(4, 3)$, is a $6-$vertex graph obtained from a cycle of length $4$ and a triangle by sharing a common vertex. Earlier it is shown that $λ(G)\leq \frac{1+\sqrt{4m-3}}{2}$ holds for all $H(4,3)-$free graphs of odd size $m\geq 44,$ and the equality holds if and only if $G\cong S_{\frac{m+3}{2},2},$ where $S_{\frac{m+3}{2},2}$ is the $m-$edge book graph $K_2 \vee \frac{m-1}{2}K_1,$ where $K_2 \vee \frac{m-1}{2}K_1,$ denotes the join of $K_2$ and $\frac{m-1}{2}K_1.$ Let $\mathcal{G}(m,H(4,3))$ denote the family of $H(4,3)$-free graphs with $m$ edges and no isolated vertices. We write $ \mathcal{G}(m,H(4,3)) \setminus \left\{ K_2 \vee \tfrac{m-1}{2}K_1 \right\} $ for the corresponding subfamily obtained by excluding the book graph. In this paper, we establish a sharp upper bound on the spectral radius of graphs over $\mathcal{G}(m,H(4,3))\setminus \{K_2 \vee \frac{m-1}{2}K_1\}$ for odd $m\geq 58$ and characterize the unique extremal graph attaining this bound.

The spectral extrema of graphs of odd size forbidding $H(4,3)$ beyond the book graph

TL;DR

This paper addresses the spectral extremal problem for -free graphs with an odd number of edges, restricting to graphs that exclude the book graph . It proves a sharp upper bound on the spectral radius for odd , identifying the unique extremal graph and showing equals the largest root of ; the proof combines equitable partitions, quotient matrices, and edge-relocation arguments. The work situates these findings within the broader spectral Turán program and discusses connections to recent odd/even size results and potential future directions. Overall, it provides a precise extremal graph characterization and a sharp spectral bound under a natural forbidden-subgraph constraint, advancing understanding of how forbidding shapes spectral properties.

Abstract

A graph is said to be -free if it does not contain a subgraph isomorphic to . The fish graph, denoted by , is a vertex graph obtained from a cycle of length and a triangle by sharing a common vertex. Earlier it is shown that holds for all free graphs of odd size and the equality holds if and only if where is the edge book graph where denotes the join of and Let denote the family of -free graphs with edges and no isolated vertices. We write for the corresponding subfamily obtained by excluding the book graph. In this paper, we establish a sharp upper bound on the spectral radius of graphs over for odd and characterize the unique extremal graph attaining this bound.
Paper Structure (4 sections, 12 theorems, 43 equations, 1 figure)

This paper contains 4 sections, 12 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem
  4. Concluding remarks.

Key Result

Theorem 1.1

Let $G \in \mathcal{G}(m,H(4,3))\setminus\left\{K_2 \vee \frac{m-1}{2}K_1\right\}$ with odd size $m \ge 58$. Then $\lambda(G) \le \tilde{\lambda}(m),$ with equality if and only if $G \cong K_1 \vee \left( K_{1,\frac{m-3}{2}} \cup 2K_1 \right),$ where $\tilde{\lambda}(m)$ is the largest root of $x^4

Figures (1)

  • Figure :

Theorems & Definitions (12)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 2 more