A short proof of Haemers' conjecture on the Seidel energy of graphs (Q6544511)

Let \(G\) be a finite simple graph with the vertex set \(V (G) = \{v_1,v_2,\ldots, v_n\}\). The Seidel matrix of \(G\), denoted by \(S(G)\), is a square matrix of order \(n\) whose diagonal entries are zero; the \((i,j)\)-th entry is \(-1\) if \(v_i\) and \(v_j\) are adjacent, and otherwise, it is 1. The Seidel energy of \(G\), denoted by \(\mathcal{E}(S(G))\), is defined to be the sum of absolute values of the eigenvalues of \(S(G)\). Then the Seidel energy of the complete graph \(K_n\) is \(2n-2\). \textit{W. H. Haemers} [MATCH Commun. Math. Comput. Chem. 68, No. 3, 653--659 (2012; Zbl 1289.05290)] conjectured that among all \(n\) vertex simple graphs, \(K_n\) has the least energy. In other words, if \(G\) is a simple graph of order \(n\), then \N\[\N\mathcal{E}(S(G))\geq\mathcal{E}(S(K_n))=2n-2.\N\]\N\textit{S. Akbari} et al. [Eur. J. Comb. 86, Article ID 103078, 8 p. (2020; Zbl 1437.05130)] proved this conjecture. This paper provides an alternative and relatively simple proof of the conjecture.