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Let $G$ be a triangle-free graph on $n$ vertices with adjacency matrixeigenvalues $\mu_1(G)\geq \mu_2(G)\geq \dots \geq \mu_n(G)$. In this paper westudy the quantity $$\mu_1(G)+\mu_n(G).$$ We prove that for any triangle-freegraph $G$ we have $$\mu_1(G)+\mu_n(G)\leq (3-2\sqrt{2})n.$$ This was proved forregular graphs by Brandt, we show that the condition on regularity is notnecessary. We also prove that among triangle-free strongly regular graphs theHigman-Sims graph achieves the maximum of $$\frac{\mu_1(G)+\mu_n(G)}{n}.$$

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