Tian and Meng in [Y. Tian and J. Meng, c  -Optimally half vertex transitive graphs with regularity , Information Processing Letters 109 (2009) 683-686] shown that a connected half vertex transitive graph with regularity and girth is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree k k   6 g G   ...

the vertex-edge wiener index of a simple connected graph g is defined as the sum of distances between vertices and edges of g. two possible distances d_1(u,e|g) and d_2(u,e|g) between a vertex u and an edge e of g were considered in the literature and according to them, the corresponding vertex-edge wiener indices w_{ve_1}(g) and w_{ve_2}(g) were introduced. in this paper, we present exact form...

A graph G is called super edge-magic if there exists a bijective function f : V (G) ∪ E (G) → {1, 2, . . . , |V (G)|+ |E (G)|} such that f (V (G)) = {1, 2, . . . , |V (G)|} and f (u) + f (v) + f (uv) is a constant for each uv ∈ E (G). A graph G with isolated vertices is called pseudo super edge-magic if there exists a bijective function f : V (G) → {1, 2, . . . , |V (G)|} such that the set {f (...

graceful labeling of G. Such sets P and Q are called the vertex values set and edge labels set of G, respectively. The notions of super-edge-graceful graphs and edge-graceful graphs are different. Mitchem and Simoson [6] showed that the step graph C2 6 is super-edge-graceful but not edge-graceful. Shiu [7] showed that the complete graph K4 is edge-graceful but not super-edge-graceful. Mitchem a...

In this paper we introduce the concept of perfect super edge-magic graphs and we prove some classes of graphs to be perfect super edge-magic.

Let G = (V,E) be a graph of order p and size q. It is known that if G is super edge-magic graph then q ≤ 2p− 3. Furthermore, if G is super edge-magic and q = 2p− 3, then the girth of G is 3. It is also known that if the girth of G is at least 4 and G is super edge-magic then q ≤ 2p − 5. In this paper we show that there are infinitely many graphs which are super edge-magic, have girth 5, and q =...

Let G = (V,E) be a connected graph. G is said to be super edge connected (or super-k for short) if every minimum edge cut of G isolates one of the vertex of G. A graph G is called m-super-k if for any edge set S # E(G) with jSj 6m, G S is still super-k. The maximum cardinality of m-super-k is called the edge fault tolerance of super edge connectivity of G. In this paper, we discuss the edge fau...

The super connectivity κ ′ and the super edge-connectivity λ′ are more refined network reliability indices than connectivity κ and edge-connectivity λ. This paper shows that for a connected graph G with order at least four rather than a star and its line graph L(G), κ ′(L(G))= λ′(G) if and only if G is not super-λ′. As a consequence, we obtain the result of Hellwig et al. [Note on the connectiv...

Let G be a connected graph of order n, minimum degree δ(G), and edge-connectivity κ (G). The graph G ismaximally edge-connected if κ (G) = δ(G) and super edge-connected if every minimum edge-cut consists of edges incident with a vertex of minimum degree. A list (d1, . . . , dn) is graphic if there is a graph with vertices v1, . . . , vn such that d(vi) = di for 1 ≤ i ≤ n. A graphic list D is su...