Let G be a simple, connected graph and let A(G) the adjacency matrix of G. If D(G) is diagonal vertex degrees G, then for every real $$\alpha \in [0,1]$$ , $$A_{\alpha }(G)$$ defined as }(G) = \alpha + (1- ) A(G).$$ The eigenvalues form }$$ -spectrum $$G_1 {\dot{\vee }} G_2$$ {\underline{\vee \langle \text {v} \rangle {e} denote subdivision-vertex join, subdivision-edge R-vertex join R-edge two...

A graph $G$ is called $P_4$-free, if $G$ does not contain an induced subgraph $P_4$. The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x)=sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,x)$ is called a domination root of $G$. In this paper we state and prove formula for the domination polynomial of no...

Let G be a simple connected graph. The first and second Zagreb indices have been introduced as  vV(G) (v)2 M1(G) degG and M2(G)  uvE(G)degG(u)degG(v) , respectively, where degG v(degG u) is the degree of vertex v (u) . In this paper, we define a new distance-based named HyperZagreb as e uv E(G) . (v))2 HM(G)     (degG(u)  degG In this paper, the HyperZagreb index of the Cartesian p...

Let $\Gamma$ be a $d$-bounded distance-regular graph with $d\geq 3$. Suppose that $P(x)$ is a set of all strongly closed subgraphs containing $x$ and that $P(x,i)$ is a subset of $P(x)$ consisting of the elements of $P(x)$ with diameter $i$. Let ${\mathcal{L}}'(x,i)$ be the set generated by the join of the elements in $P(x,i)$. By ordering ${\mathcal{L}}'(x,i)$ by inclusion or reverse inclusion...

A finite abstract simplicial complex G defines two finite simple graphs: the Barycentric refinement G1, connecting two simplices if one is a subset of the other and the connection graph G′, connecting two simplices if they intersect. We prove that the Poincaré-Hopf value i(x) = 1−χ(S(x)), where χ(S(x)) is the Euler characteristics of the unit sphere S(x) of a vertex x in G1, agrees with the Gre...

We consider only finite undirected graphs without loops or multiple edges. Let G be a graph with vertex set V(G) and edge set E(G). For a vertex x ∈V(G), we write NG(v) for the set of vertices of V(G) adjacent to v, NG[v] for NG(v) ⋃{v}, and dG(v) = |NG(v)| for the degree of v in G. If S and T are disjoint subsets of V(G), then eG(S,T) denotes the number of edges that join S and T , and G− S de...