Extremal $Q$-index problem in outerplanar graphs

TL;DR

This paper investigates the maximum signless Laplacian spectral index in outerplanar graphs under certain forbidden subgraph conditions, identifying unique extremal graphs for these cases.

Contribution

It determines the unique extremal outerplanar graphs that maximize the $Q$-index when forbidding specific cycles or disjoint unions of paths.

Findings

Identifies the extremal graphs for forbidden cycles.

Determines the extremal graphs for disjoint unions of paths.

Provides exact maximum $Q$-index values for these classes.

Abstract

Outerplanar Tur\'an 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.