Note on the sum of the smallest and largest eigenvalues of a triangle-free graph

Let G be a triangle-free graph on n vertices with adjacency matrix eigenvalues μ1(G)≥μ2(G)≥…≥μn(G). In this paper we study the quantityμ1(G)+μn(G). We prove that for any triangle-free graph G we haveμ1(G)+μn(G)≤(3−22)n. This was proved for regular graphs by Brandt, we show that the condition on regularity is not necessary. We also prove that among triangle-free strongly regular graphs the Higman-Sims graph achieves the maximum ofμ1(G)+μn(G)n.