A lower bound for the sum of the two largest signless Laplacian eigenvalues

Research paper by Carla Silva Oliveira, Leonardo de Lima

Indexed on: 19 Nov '16Published on: 17 Nov '16Published in: Electronic Notes in Discrete Mathematics


Let G   be a connected graph of order n≥3n≥3 and let Q(G)=D(G)+A(G)Q(G)=D(G)+A(G) be the signless Laplacian of G  , where A(G)A(G) is the adjacency matrix and D(G)D(G) is the diagonal matrix of the row-sums of A(G)A(G). Write q1(G)q1(G) and q2(G)q2(G) for the two largest eigenvalues of Q(G)Q(G). In this paper, we obtain a lower bound to the sum of the two Q  –largest eigenvalues, that is, q1(G)+q2(G)≥d1(G)+d2(G)+1q1(G)+q2(G)≥d1(G)+d2(G)+1 with equality if and only if G   is the star SnSn or the complete graph K3K3, where didi is the i–largest degree of a vertex of G.