k-Restricted edge connectivity is an important parameter in measuring the reliability and fault tolerance of large interconnection networks. In this paper we present two families of graphs similar with the networks proposed by Chen et al. [Y.C. Chen, J.J.M. Tan, L.H. Hsu, S.S. Kao, Super-connectivity and super edge-connectivity for some interconnection networks, Applied Mathematics and Computat...

Let G be a (multi)graph of order n and let u,v vertices G. The maximum number internally disjoint u–v paths in is denoted by κG(u,v), the edge-disjoint λG(u,v). average connectivity defined κ¯(G)=∑κG(u,v)∕n2, edge-connectivity λ¯(G)=∑λG(u,v)∕n2, where both sums run over all unordered pairs {u,v}⊆V(G). A graph called ideally connected if κG(u,v)=min{deg(u),deg(v)} for {u,v} We prove that every m...

A graph G is called edge-magic if there exists a bijective function φ : V (G)∪E(G) → {1, 2,. .. , |V (G)|+ |E(G)|} such that φ(x)+φ(xy)+φ(y) is a constant c(φ) for every edge xy ∈ E(G); here c(φ) is called the valence of φ. A graph G is said to be super edge-magic if φ(V (G)) = {1, 2,. .. , |V (G)|}. The super edge-magic deficiency, denoted by μ s (G), is the minimum nonnegative integer n such ...

We focus on the algorithm underlying the main result of [6]. This is an algebraic formula to generate all connected graphs in a recursive and efficient manner. The key feature is that each graph carries a scalar factor given by the inverse of the order of its group of automorphisms. In the present paper, we revise that algorithm on the level of graphs. Moreover, we extend the result subsequentl...

The revised edge-Szeged index of a connected graph $G$ is defined as Sze*(G)=∑e=uv∊E(G)( (mu(e|G)+(m0(e|G)/2)(mv(e|G)+(m0(e|G)/2) ), where mu(e|G), mv(e|G) and m0(e|G) are, respectively, the number of edges of G lying closer to vertex u than to vertex v, the number of ed...

The Kronecker product of two connected graphs G1,G2, denoted by G1 × G2, is the graph with vertex set V (G1 ×G2) = V (G1)×V (G2) and edge set E(G1 ×G2) = {(u1, v1)(u2, v2) : u1u2 ∈ E(G1), v1v2 ∈ E(G2)}. The kth power Gk of G is the graph with vertex set V (G) such that two distinct vertices are adjacent in Gk if and only if their distance apart in G is at most k. A connected graph G is called s...