The domination number and the least QQ-eigenvalue

A vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all dominating sets of $G$. For a graph, the least $Q$-eigenvalue is the least eigenvalue of its signless Laplacian matrix. In this paper, for a nonbipartite graph with both order $n$ and domination number $\gamma$, we show that $n\geq 3\gamma-1$, and show that it contains a unicyclic spanning subgraph with the same domination number $\gamma$. By investigating the relation between the domination number and the least $Q$-eigenvalue of a graph, we minimize the least $Q$-eigenvalue among all the nonbipartite graphs with given domination number.Comment: 13 pages, 3 figure

Signless Laplacian spectral radii of graphs with given chromatic number

AbstractLet G be a simple graph with vertices v1,v2,…,vn, of degrees Δ=d1⩾d2⩾⋯⩾dn=δ, respectively. Let A be the (0,1)-adjacency matrix of G and D be the diagonal matrix diag(d1,d2,…,dn). Q(G)=D+A is called the signless Laplacian of G. The largest eigenvalue of Q(G) is called the signless Laplacian spectral radius or Q-spectral radius of G. Denote by χ(G) the chromatic number for a graph G. In this paper, for graphs with order n, the extremal graphs with both the given chromatic number and the maximal Q-spectral radius are characterized, the extremal graphs with both the given chromatic number χ≠4,5,6,7 and the minimal Q-spectral radius are characterized as well