Throughout this section, G = (V,E) is a multi-graph. A cut of G is a bipartition (S, S̄) of the vertex set of G. The capacity of a cut is the number of edges having one endpoint on both sides of the cut. A min-cut is a cut of minimum capacity. A minimum cut can be computed with the help of maxflow computations. For some vertex s and every other vertex t, one computes the minimum cut separating s...

Suppose a finitely generated group [Formula: see text] is hyperbolic relative to set of proper subgroups text]. Established results in the literature imply that “visual” metric on “linearly connected” if and only boundary has no cut point. Our goal produce linearly connected metrics are “piecewise” visual when contains points. main theorem graph groups decompositions relatively by work B. Bowdi...

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...

The emph{Harary index} $H(G)$ of a connected graph $G$ is defined as $H(G)=sum_{u,vin V(G)}frac{1}{d_G(u,v)}$ where $d_G(u,v)$ is the distance between vertices $u$ and $v$ of $G$. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ ...

‎an  oriented perfect path double cover (oppdc) of a‎ ‎graph $g$ is a collection of directed paths in the symmetric‎ ‎orientation $g_s$ of‎ ‎$g$ such that‎ ‎each arc‎ ‎of $g_s$ lies in exactly one of the paths and each‎ ‎vertex of $g$ appears just once as a beginning and just once as an‎ ‎end of a path‎. ‎maxov{'a} and ne{v{s}}et{v{r}}il (discrete‎ ‎math‎. ‎276 (2004) 287-294) conjectured that ...

let $g$ be an $(n,m)$-graph. we say that $g$ has property $(ast)$if for every pair of its adjacent vertices $x$ and $y$, thereexists a vertex $z$, such that $z$ is not adjacentto either $x$ or $y$. if the graph $g$ has property $(ast)$, thenits complement $overline g$ is connected, has diameter 2, and itswiener index is equal to $binom{n}{2}+m$, i.e., the wiener indexis insensitive of any other...

In this paper, we study the problem of fast dynamic pointer following: given a directed graph G where each vertex has outdegree 1, efficiently support the operations of i) changing the outgoing edge of any vertex, and ii) find the vertex k vertices ‘after’ a given vertex. We exhibit a solution to this problem based on link-cut trees that requires O(lgn) time per operation, and prove that this i...

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...

For any vertex v and any edge e in a non-trivial connected graph G, the eccentricity e(v) of v is e(v) =max{d(v, u) : u ∈ V }, the vertex-to-edge eccentricity e1(v) of v is e1(v) = max{d(v, e) : e ∈ E}, the edge-to-vertex eccentricity e2(e) of e is e2(e) = max{d(e, u) : u ∈ V } and the edge-to-edge eccentricity e3(e) of e is e3(e) = max{d(e, f) : f ∈ E}. The set C(G) of all vertices v for which...

For an edge-colored graph G, we call edge-cut M of G monochromatic if the edges are colored with same color. The is called disconnected any two distinct vertices separated by a edge-cut. connected disconnection number (or MD-number for short) denoted md(G), maximum colors that allowed in order to make disconnected. graphs diameter one, they complete and so their MD-numbers 1. at least 3, can co...