A shorter proof of the distance energy of complete multipartite graphs

Abstract

Caporossi, Chasser and Furtula in [Les Cahiers du GERAD (2009) G-2009 conjectured that the distance energy of a complete multipartite graph of order n with r ≥ parts, each of size at least 2, is equal to (n − r). Stevanovic, Milosevic, Hic and Pokorny in [MATCH Commun. Math. Comput. Chem. 70 (2013), no. 1, 157-162.] proved the conjecture, and then Zhang in [Linear Algebra Appl. 450 (2014), 108-120.] gave another proof. We give a shorter proof of this conjecture using the interlacing inequalities of a positve semi-de nite rank-1 perturbation to a real symmetric matrix.

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