Connected (K<inf>4</inf>−e)-free graphs whose second largest eigenvalue does not exceed 1

Abstract

Connected graphs whose second largest eigenvalue λ2 does not exceed 1 have been investigated in the last four decades. Over the years only few particular classes with this spectral property are completely determined. For example, they include bipartite graphs, line graphs and threshold graphs. In this paper we determine all connected (K4−e)-free graphs, i.e., the graphs that do not contain an induced subgraph isomorphic to the graph obtained by removing an edge from the complete graph of order 4; such graphs are also called diamond-free graphs. Since every triangle-free graph is automatically diamond-free, our results includes all triangle-free graphs with λ2≤1. Moreover, it includes all connected bipartite graphs with the same spectral property, and therefore strengthens the result of Petrović (J. Combin. Theory B, 1991).