A characterization of graphs with 3-path coverings and the evaluation of the minimum 3-covering energy of complete graphs and star graph with \(m\) rays of length 2 (Q486879)

A 3-covering of a graph \(G\) is a set of vertices \(Q\) such that each three-vertex path in \(G\) contains a vertex in \(Q\). The author forms the 3-covering matrix of \(G\) from its adjacency matrix by additionally setting to 1 all the diagonal entries that correspond to vertices in a given 3-covering. After defining the 3-covering energy as the sum of absolute values of the eigenvalues of this 3-covering matrix, the author proceeds to find the value of this invariant for a few very specific graph types.