The Signless Laplacian Spectral Radius With Given Independence Number

The theory of spectra is an important component of graph theory. It comprises adjacent spectrum, Laplacian spectrum, signless Laplacian spectrum,etc. The signless Laplacian spec-trum is more closely relative to the graph structures than other spectrum. The independence numberα(G) of G is defined as the maximum cardinality of a set of pairwise non-adjacent vertices which is called an independent set. A connected graph is said to be bicyclic, if |E(G)|=|V(G)|+1.Let B(n, a) be the class of bicyclic graphs on n vertices with indepen-dence numberα,B1 (n, a) be the subclass of B(n, a) consisting of all bicyclic graphs with two edge-disjoint cycles and B2(n, a)=B(n, a)\B1(n, a). In this paper, we use signless Lapla-cian spectrum to study the structures and properties of graph with given independence number. Firstly, we give the results about the bounds of signless Laplacian spectra and the extremal graph of the spectral radius with given constraints. Secondly, we characterize the graphs which have the minimum spectral radius among all the connected graphs of order n with independence numberα(G)∈{1,2, [n/2], [n/2]+1,n-2,n-3,n-1}. At last, we determines the unique graph with the maximal signless Laplacian spectral radius among all bicyclic graphs in B1(n, a) and B2(n, a), respectively. Furthermore, the upper bound of the signless Laplacian spectral radius and the extremal graph for all bicyclic graphs B(n, a) are also obtained.