The clique chromatic number of a graph is the minimum colours needed to colour its vertices so that no inclusion-wise maximal which not an isolated vertex monochromatic. We show every maximum degree $\Delta$ has $O\left(\frac{\Delta}{\log~\Delta}\right)$. obtain as corollary $n$-vertex $O\left(\sqrt{\frac{n}{\log ~n}}\right)$. Both these results are tight.

Let $G$ be a non-abelian group and let $Z(G)$ be the center of $G$. Associate with $G$ there is agraph $Gamma_G$ as follows: Take $Gsetminus Z(G)$ as vertices of$Gamma_G$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $Gamma_G$ is called the non-commuting graph of $G$. In recent years many interesting works have been done in non-commutative graph of groups. Computing the clique...

Let $G$ be a graph with vertex set $V(G)$. For any integer $kge 1$, a signed (total) $k$-dominating functionis a function $f: V(G) rightarrow { -1, 1}$ satisfying $sum_{xin N[v]}f(x)ge k$ ($sum_{xin N(v)}f(x)ge k$)for every $vin V(G)$, where $N(v)$ is the neighborhood of $v$ and $N[v]=N(v)cup{v}$. The minimum of the values$sum_{vin V(G)}f(v)$, taken over all signed (total) $k$-dominating functi...

For a set of graphs F, let ex(n,F) and spex(n,F) denote the maximum number edges spectral radius an n-vertex F-free graph, respectively. Nikiforov (LAA, 2007) gave version Turán Theorem by showing that spex(n,Kk+1)=λ(Tk(n)), where Tk(n) is k-partite graph on n vertices. In same year, Feng, Yu Zhang (LAA) determined exact value spex(n,Ms+1), Ms+1 matching with s+1 edges. Recently, Alon Frankl (a...

Let G be a simple graph with vertex set {v1, v2, … , vn}. The common neighborhood graph of G, denoted by con(G), is a graph with vertex set {v1, v2, … , vn}, in which two vertices are adjacent if and only if they have at least one common neighbor in the graph G. In this paper, we compute the common neighborhood of some composite graphs. In continue, we investigate the relation between hamiltoni...

Since χ(G) · α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that any graph G has the complete graph Kdn α e as a minor, where n is the number of vertices of G and α is the maximum number of independent vertices in G. Motivated by this fact, it is shown that any graph on n vertices with independence number α ≥ 3 has the complete graph Kd n 2α−2 e as a minor. This improves the well-known theorem o...