A Study On Q-integral Graphs

Graphs that are discussed in this paper are simple connected(except it has specific ex-planation), i.e. finite undirected connected graphs without loops or multiple edges. We useA(G), L(G) and Q(G) to denote adjacency matrix, Laplacian matrix and Signless Laplacianmatrix of a graph G, respectively, where L(G) = D(G)-A(G), Q(G) = D(G)+A(G) andD(G) is the diagonal matrix of vertex degrees of a graph G. The quest for integral graphs[15]was initiated by F. Harary and A. J. Schwenk in 1973. Then, a large number of lovers ofmathematics are interested in this problem, and the study of integral graphs is extendedto the study of L-integral graphs and Q-integral graphs. Over the years, the integral andL-integral graphs are well studied. On the other hand, the graphs with integral Q-spectrumare hardly studied , so far there are exactly two forthcoming paper[13,14]. Therefore, wewill investigate the study of Q-integral graphs. This paper is divided into four parts, asfollowing,In the first part, we introduce the general konwledge, domestic and foreign researchingsituations.In the second part, we study the connected graphs with 6 edge-regular Q-integralgraphs avoiding 1 and 7 in the Q-spectrum. This work is studied in three cases here.Firstly, the graph is (r,s)-semiregular bipartite graph (r + s = 8 and r < s). Secondly, thegraph is 4-regular bipartite graph. Lastly, the graph is 4-regular nonbipartite graph, and thecorresponding results are given respectively.In the third part, we study the relation connecting the Q-polynomial of a graph G andits complement graph (G|—). This study here consists of two aspects. Firstly, we obtain a relationconnecting the Q-polynomial of a regular graph and its complement graph. Secondly, wealso obtain a relation connecting the Q-polynomial of the complete product of two graphsand these two graphs. The main results are made respectively.In the forth part, we mainly study the Q-integral tree and the line graph of trees isintegral.