A tight $Q$-index condition for a graph to be $k$-path-coverable involving minimum degree

TL;DR

This paper establishes a precise spectral condition involving the $Q$-index that guarantees a large, connected graph with fixed minimum degree can be covered by at most $k$ disjoint paths.

Contribution

It introduces a new tight sufficient condition based on the $Q$-index for $k$-path-coverability in graphs with fixed minimum degree.

Findings

Provides a tight spectral condition for $k$-path-coverability.

Applicable to large, connected graphs with fixed minimum degree.

Enhances understanding of spectral graph theory related to path covers.

Abstract

A graph $G$ is $k$-path-coverable if its vertex set $V(G)$ can be covered by $k$ or fewer vertex disjoint paths. In this paper, using the $Q$-index of a connected graph $G$, we present a tight sufficient condition for $G$ with fixed minimum degree and large order to be $k$-path-coverable.