A characterization of graphs with 3-coverings and the evaluation of the 3-covering energy of star graphs with m rays of length 2

TL;DR

This paper characterizes graphs with 3-coverings, explores their properties, and calculates the minimum 3-covering energy specifically for star graphs with m rays of length 2.

Contribution

It provides a characterization of graphs with 3-coverings and evaluates the 3-covering energy for a specific class of star graphs, extending spectral graph theory.

Findings

Characterization of graphs with 3-coverings in terms of non-Q-covered edges

Determination of minimum 3-covering energy for star graphs with m rays of length 2

Method to find eigenvalues using loops of weight 1 attached to vertices

Abstract

The smallest set Q of vertices of a graph G, such that every path on 3 vertices, has at least one vertex in Q, is a minimum 3-covering of G. By attaching loops of weight 1 to the vertices of G we can find the eigenvalues associated with G, and hence the minimum 3-covering energy of G. In this paper we characterize graphs with 3-coverings in terms of non-Q-covered edges, and we determine the minimum 3-covering energy of a star graph with m rays each of length 2.