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...

Let G be a connected graph with minimum degree id="M3"> δ and vertex-connectivity id="M4"> κ . The id="M5"> is id="M6"> k -connected if id="M7"> ≥ , maximally id="M8"> = super-connected every vertex-cut isol...

By applying the Tutte decomposition of 2{connected graphs into 3{block trees we provide unique structural characterizations of several classes of 2{connected graphs, including minimally 2{connected graphs, minimally 2{edge{connected graphs, critically 2{connected graphs, critically 2{edge{connected graphs, 3{edge{connected graphs, 2{connected cubic graphs and 3{connected cubic graphs. We also 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...

The edge-connectivity of a graph G can be defined as λ(G) = min{λG(u, v) |u, v ∈ V (G)}, where λG(u, v) is the local edge-connectivity of two vertices u and v in G. We call a graph G maximally edgeconnected when λ(G) = δ(G) and maximally local edge-connected when λG(u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. In 2000, Fricke, Oellermann and Swart (unpublished manusc...

Numerous networks as, for example, road networks, electrical networks and communication networks can be modeled by a graph. Many attempts have been made to determine how well such a network is "connected" or stated differently how much effort is required to break down communication in the system between at least some nodes. Two well-known measures that indicate how "reliable" a graph is are the...