TY - JOUR
T1 - Q-integral complete r-partite graphs
AU - Zhao, Guopeng
AU - Wang, Ligong
AU - Li, Ke
PY - 2013/2/1
Y1 - 2013/2/1
N2 - For a graph G of order n, the signless Laplacian matrix of G is Q(G)=D(G)+A(G), where A(G) is its adjacency matrix and D(G) is the diagonal matrix of the vertex degrees in G. The signless Laplacian characteristic polynomial (or Q-polynomial) of G is QG(x)=|x In-Q(G)|, where In is the n×n identity matrix. A graph G is called Q-integral if all the eigenvalues of its signless Laplacian characteristic polynomial QG(x) are integers. In this paper, we give a sufficient and necessary condition for complete r-partite graphs to be Q-integral, from which we construct infinitely many new classes of Q-integral graphs. Finally, we propose two basic open problems for further study.
AB - For a graph G of order n, the signless Laplacian matrix of G is Q(G)=D(G)+A(G), where A(G) is its adjacency matrix and D(G) is the diagonal matrix of the vertex degrees in G. The signless Laplacian characteristic polynomial (or Q-polynomial) of G is QG(x)=|x In-Q(G)|, where In is the n×n identity matrix. A graph G is called Q-integral if all the eigenvalues of its signless Laplacian characteristic polynomial QG(x) are integers. In this paper, we give a sufficient and necessary condition for complete r-partite graphs to be Q-integral, from which we construct infinitely many new classes of Q-integral graphs. Finally, we propose two basic open problems for further study.
KW - Complete r-partite graph
KW - Graph spectrum
KW - Q-integral
KW - Signless Laplacian matrix
UR - http://www.scopus.com/inward/record.url?scp=84870381578&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2012.09.009
DO - 10.1016/j.laa.2012.09.009
M3 - 文章
AN - SCOPUS:84870381578
SN - 0024-3795
VL - 438
SP - 1067
EP - 1077
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 3
ER -