Explicit expressions of the restricted edge connectivity of the Cartesian product of regular graphs are presented; some sufficient conditions for regular Cartesian product graphs to be maximally or super restricted edge connected are obtained as a result.

Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π...

Let G = (V, E) be a connected graph. Let G = (V, E) be a connected graph. An edge set F ⊂ E is said to be a k-restricted edge cut, if G− F is disconnected and every component of G− F has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is the cardinality of a minimum k-restricted edge cut of G. A graph G is λk-connected, if G contains a k-restricted edge cut. A λk...

The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second ve...

The super (resp., edge-) connectivity of a connected graph is the minimum cardinality of a vertex-cut (resp., an edge-cut) whose removal does not isolate a vertex. In this paper, we consider the two parameters for a special class of graphs G(G0, G1; M), proposed by Chen et al [Applied Math. and Computation, 140 (2003), 245-254], obtained from two k-regular k-connected graphs G0 and G1 with the ...

Preface Graphs are often used as a model for networks, for example transport networks, telecommunication systems, a network of servers and so on. Typically, the networks and thus the representing graphs are connected; that means, that there exists a path between all pairs of two vertices in the graph. But sometimes an element of the network fails, for the graphs that means the removal of a vert...

Let G=(V; E) be a connected graph and S ⊂E. S is said to be a m-restricted edge cut (m-RC) if G − S is disconnected and each component contains at least m vertices. The m-restricted edge connectivity (G) is the minimum size of all m-RCs in G. Based on the fact that (G)6 3(G), where m(G)=min{!(X ): X ⊂V; |X |=m and G[X ] is connected} (!(X ) denotes the number of edges with one end vertex in X a...

the first ($pi_1$) and the second $(pi_2$) multiplicative zagreb indices of a connected graph $g$, with vertex set $v(g)$ and edge set $e(g)$, are defined as $pi_1(g) = prod_{u in v(g)} {d_u}^2$ and $pi_2(g) = prod_{uv in e(g)} {d_u}d_{v}$, respectively, where ${d_u}$ denotes the degree of the vertex $u$. in this paper we present a simple approach to order these indices for connected graphs on ...