Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extremal $Q$-index problem in outerplanar graphs

Jin Cai, Leyou Xu, Bo Zhou

TL;DR

This work studies the maximum Q-index $q(G)$ among connected outerplanar graphs on $n$ vertices that forbid either a fixed cycle $C_\ell$ or a disjoint union of equal-length paths $tP_k$. A structural theorem shows that extremal graphs possess a dominating vertex $u$ with $G[N(u)]$ a union of paths, enabling exact identification of the extremal graphs. For cycle-forbidden cases the unique extremal graph is $K_1\vee(\alpha P_{\ell-2}\cup P_r)$ with $\alpha=\left\lfloor\frac{n-1}{\ell-2}\right\rfloor$ and $r=n-1-\alpha(\ell-2)$; for path-forbidden cases, the extremals are given explicitly by $K_1\vee (P_{t\ell-\ell-1}\cup\alpha P_{\ell-1}\cup P_r)$ (and a special $t=1$ variant) with $\alpha$ and $r$ determined by $n$, $t$, and $\ell$. The proofs combine Perron-vector methods, Rayleigh quotients, and a suite of edge-move lemmas to compare $q(G)$ under structural modifications, within the outerplanar constraints. The results complete the Q-index Turán-type extremal picture for outerplanar graphs under these two forbidden-family settings, yielding unique extremal graphs and sharpening spectral extremal graph theory in outerplanar settings.

Abstract

Outerplanar Turán problem has received considerable attention recently. We study the spectral version via $Q$-index. We determine the unique graph that maximizes the $Q$-index among all $n$-vertex connected outerplanar graphs which are respectively forbidden to contain: (i) a fixed cycle; and (ii) the disjoint union of paths of a given order.

Extremal $Q$-index problem in outerplanar graphs

TL;DR

This work studies the maximum Q-index among connected outerplanar graphs on vertices that forbid either a fixed cycle or a disjoint union of equal-length paths . A structural theorem shows that extremal graphs possess a dominating vertex with a union of paths, enabling exact identification of the extremal graphs. For cycle-forbidden cases the unique extremal graph is with and ; for path-forbidden cases, the extremals are given explicitly by (and a special variant) with and determined by , , and . The proofs combine Perron-vector methods, Rayleigh quotients, and a suite of edge-move lemmas to compare under structural modifications, within the outerplanar constraints. The results complete the Q-index Turán-type extremal picture for outerplanar graphs under these two forbidden-family settings, yielding unique extremal graphs and sharpening spectral extremal graph theory in outerplanar settings.

Abstract

Outerplanar Turán problem has received considerable attention recently. We study the spectral version via -index. We determine the unique graph that maximizes the -index among all -vertex connected outerplanar graphs which are respectively forbidden to contain: (i) a fixed cycle; and (ii) the disjoint union of paths of a given order.
Paper Structure (4 sections, 14 theorems, 31 equations)

This paper contains 4 sections, 14 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['str']}
  4. Proof of Theorems \ref{['cycle']} and \ref{['path']}

Key Result

Theorem 1

Let $G$ be a connected outerplanar $C_\ell$-free graph of order $n$ with $\ell\ge 3$. Then $q(G)\le q(K_1\vee (\alpha P_{\ell-2}\cup P_{r}))$ with equality if and only if $G\cong K_1\vee (\alpha P_{\ell-2}\cup P_{r})$, where $\alpha =\lfloor\frac{n-1}{\ell-2}\rfloor$ and $r=n-1-\alpha(\ell-2)$.

Theorems & Definitions (28)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • proof
  • ...and 18 more