Spectral conditions implying the existence of doubly chorded cycles without or with constraints

TL;DR

The paper addresses spectral conditions that force the existence of chorded cycles, focusing on doubly chorded cycles (DCC) and DCCs with two chords incident to a vertex (DCC$_1$). It develops tight bounds on the spectral radius $\rho(G)$ and fully characterizes extremal, chordless structures via Perron-Frobenius theory, equitable quotient matrices, and detailed neighborhood analyses around a Perron vertex. The main results give a sharp bound $\rho(G)\le \frac{1}{2}+\sqrt{2n-\frac{15}{4}}$ for graphs with no DCC (equality at $K_{1,1,n-2}$) and a complete, piecewise extremal classification for graphs with no DCC$_1$ (equality at $F_1$ for $n=5$, at $K_{1,1,n-2}$ for $6\le n\le 9$, and at $K_{3,n-3}$ for $n\ge 10$). These results advance Brualdi–Solheim–Turán type spectral questions for chorded cycles and connect to broader spectral-extremal problems in graph theory. The work provides exact spectral thresholds and explicit extremal graphs, enriching the understanding of how spectral data governs the presence or absence of complex cycle structures.

Abstract

What spectral conditions imply a graph contains a chorded cycle? This question was asked by R.J. Gould in 2022. We answer two modified versions of Gould's question by giving tight spectral conditions that imply the existence of doubly chorded cycle, and a doubly chorded cycle with two chords incident to a vertex, respectively.

Theorems & Definitions (57)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1
• proof