Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs. | On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs Saieed Akbari1 2 Ebrahim Ghorbani1 2 Jacobus H. Koolen3 4 Mohammad Reza Oboudi1 2 department of Mathematical Sciences Sharif University of Technology . Box 11155-9415 Tehran Iran s_akbari@ e_ghorbani@ m_r_oboudi@ 2School of Mathematics Institute for Research in Fundamental Sciences IPM . Box 19395-5746 Tehran Iran department of Mathematics Pohang University of Science and Technology POSTECH Pohang 790-785 South Korea koolen@ 4Pohang Mathematics Institute PMI Pohang University of Science and Technology POSTECH Pohang 790-785 South Korea Submitted 12 Jan 2010 Accepted 27 Jul 2010 Published 16 Aug 2010 Mathematics Subject Classifications 05C50 Abstract Let G be a graph of order n with signless Laplacian eigenvalues q1 . qn and Laplacian eigenvalues ạ1 . ụn. It is proved that for any real number a with 0 a 1 or 2 a 3 the inequality q qn ft rfa En. holds and for any real number ft with 1 ft 2 the inequality qf qn Ẹf ụỉỉi holds. In both inequalities the equality is attained for a ị 1 2 if and only if G is bipartite. 1 Introduction Let G be a graph with vertex set V G v1 . vn and edge set E G e1 . em . The adjacency matrix of G A aij is an n X n matrix such that aij 1 if vi and Vj THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R115 1 are adjacent and otherwise aij 0. The incidence matrix of G denoted by X xjj is the n X m matrix whose rows are indexed by the set of vertices of G and whose columns are indexed by the set of edges of G defined by _ i 1 if ej is incident with vp Xij 0 otherwise. If we consider an orientation for G then in a similar manner as for the incidence matrix the directed incidence matrix of the oriented graph G denoted by D dij is defined as 1 if ej is an incomming edge to vi -1 if ej is an outgoinging edge from vi 0 otherwise. Let A be the diagonal matrix whose entries are vertex degrees of G. The Laplacian matrix
