Eigenvalues and triangles in graphs

Abstract Bollobás and Nikiforov (J. Combin. Theory Ser. B. 97 (2007) 859–865) conjectured the following. If G is a Kr+1-free graph on at least r+1 vertices and m edges, then ${\rm{\lambda }}_1^2(G) + {\rm{\lambda }}_2^2(G) \le (r - 1)/r \cdot 2m$, where λ1 (G)and λ2 (G) are the largest and the second largest eigenvalues of the adjacency matrix A(G), respectively. In this paper we confirm the conjecture in the case r=2, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdös and Nosal respectively, we prove that every non-bipartite graph of order and size contains a triangle if one of the following is true: (i) ${{\rm{\lambda }}_1}(G) \ge \sqrt {m - 1} $ and $G \ne {C_5} \cup (n - 5){K_1}$, and (ii) ${{\rm{\lambda }}_1}(G) \ge {{\rm{\lambda }}_1}(S({K_{[(n - 1)/2],[(n - 1)/2]}}))$ and $G \ne S({K_{[(n - 1)/2],[(n - 1)/2]}})$, where $S({K_{[(n - 1)/2],[(n - 1)/2]}})$ is obtained from ${K_{[(n - 1)/2],[(n - 1)/2]}}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems.