let g be a simple graph with vertex set v(g) {v1,v2 ,...vn} . for every vertex i v , ( ) i  vrepresents the degree of vertex i v . the h-th order of randić index, h r is defined as the sumof terms1 2 11( ), ( )... ( ) i i ih  v  v  vover all paths of length h contained (as sub graphs) in g . inthis paper , some bounds for higher randić index and a method for computing the higherrandic ind...

Given a decomposable graph, we characterize and enumerate the set of pairs of vertices whose connection or disconnection results in a new graph that is also decomposable. We discuss the relevance of this results to Markov chain Monte Carlo methods that sample or optimize over the space of decomposable graphical models according to probabilities determined by a posterior distribution given obser...

in this paper, at first we mention to some results related to pi and vertex co-pi indices and then we introduce the edge versions of co-pi indices. then, we obtain some properties about these new indices.

The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

Let G be a simple graph with vertex set V(G) {v1,v2 ,...vn} . For every vertex i v , ( ) i  v represents the degree of vertex i v . The h-th order of Randić index, h R is defined as the sum of terms 1 2 1 1 ( ), ( )... ( ) i i ih  v  v  v  over all paths of length h contained (as sub graphs) in G . In this paper , some bounds for higher Randić index and a method for computing the higher R...

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

Let G be a simple graph with vertex set V (G). The common neighborhood graph or congraph of G, denoted by con(G), is a graph with vertex set V (G), in which two vertices are adjacent if and only if they have at least one common neighbor in G. We compute the congraphs of some composite graphs. Using these results, the congraphs of several special graphs are determined.

If G is a graph, its Laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix. The main thrust of the present article is to prove several Laplacian eigenvector “principles” which in certain cases can be used to deduce the effect on the spectrum of contracting, adding or deleting edges and/or of coalescing vertices. One application is the construction of ...

This thesis deals with the multi-terminal vertex separator problem. Given a graph G = (V ∪T,E) with V ∪T the set of vertices, where T is a set of terminals, and a weight function w : V → Z, associated with nonterminal nodes, the multi-terminal vertex separator problem consists in partitioning V ∪T into k+1 subsets {S, V1, . . . , Vk} such that there is no edge between two different subsets Vi a...