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

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