let $g=(v,e)$ be a connected graph. the eccentric connectivity index of $g$, $xi^{c}(g)$, is defined as $xi^{c}(g)=sum_{vin v(g)}deg(v)ec(v)$, where $deg(v)$ is the degree of a vertex $v$ and $ec(v)$ is its eccentricity. the eccentric distance sum of $g$ is defined as $xi^{d}(g)=sum_{vin v(g)}ec(v)d(v)$, where $d(v)=sum_{uin v(g)}d_{g}(u,v)$ and $d_{g}(u,v)$ is the distance between $u$ and $v$ ...

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

We disprove a conjecture of Frank [4] stating that each weakly 2k-connected graph has a k-vertex-connected orientation. For k ≥ 3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.

For an ordered set W = {w1, w2, · · · , wk} of vertices and a vertex v in a connected graph G, the representation of v with respect to W is the k-vector r(v|W ) = (d(v, w1), d(v, w2), · · · , d(v, wk)), where d(x, y) represents the distance between the vertices x and y. The set W is a connected resolving set for G if distinct vertices of G have distinct representations with respect to W and the...

Let G be a given connected graph on n vertices. Suppose N facilities are located in the vertices of the graph. We consider the expected distance between two randomly chosen facilities. This is modelled by the following definition: Let G be a connected graph and let each vertex have weight c(v). The average distance of G with respect to c is defined as µ c (G) = N 2 −1 {u,v}⊆V (G) d G (u, v), wh...

We show that, in a k-connected graph G of order n with α(G) = α, between any pair of vertices, there exists a path P joining them with |P | ≥ min { n, (k−1)(n−k) α + k } . This implies that, for any edge e ∈ E(G), there is a cycle containing e of length at least min { n, (k−1)(n−k) α + k } . Moreover, we generalize our result as follows: for any choice S of s ≤ k vertices in G, there exists a t...

Let G be a connected graph, where k2. S. Smith conjectured that every two longest cycles of G have at least k vertices in common. In this note, we show that every two longest cycles meet in at least ck 3Â5 vertices, where cr0.2615. 1998 Academic Press In this note, we provide a lower bound on the number of vertices in the intersection of any two longest cycles in a k-connected graph (k2). This ...

The inclusive edge (vertex, mixed) connectivity of a vertex v is the minimum number of edges (vertices, graph elements) whose removal yields a subgraph in which v is a cutvertex. Stability under edge addition, in which the value of the parameter remains unchanged with the addition of any edge, is investigated. In particular we examine relationships between stability of the inclusive vertex conn...

The augmented eccentric connectivity index of a graph which is a generalization of eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph. In this paper we established some relationships between augmented eccentric connectivity index and several other graph invariants like number of ...

Vertex connectivity and edge connectivity are two important parameters in interconnection networks. Even though they reflect the fault tolerance correctly, they undervalue the resilience of large networks. By the concept of conditional connectivity and super-connectivity, the concept of restricted vertex connectivity and restricted edge connectivity of graphs was proposed by Esfahanian [A.H. Es...