This work deals with a generalization of the Cartesian product of graphs, the product graph Gm ∗Gp introduced by Bermond et al. [J.C. Bermond, C. Delorme, G. Farhi, Large graphs with givendegree anddiameter II, J. Combin. Theory, Series B 36 (1984), 32–48]. The connectivity of these product graphs is approached by studying the k-restricted edge-connectivity, which is defined as the minimum numb...

For an integer s > 0 and for u, v ∈ V (G) with u ≠ v, an (s; u, v)-trail-system of G is a subgraphH consisting of s edge-disjoint (u, v)-trails. A graph is supereulerianwithwidth s if for any u, v ∈ V (G) with u ≠ v, G has a spanning (s; u, v)-trail-system. The supereulerian width μ(G) of a graph G is the largest integer s such that G is supereulerian with width k for every integer kwith 0 ≤ k ...

Abstract. Let G be a nontrivial connected graph on which is defined a coloring N k k G E c ∈ → }, ,...., 3 , 2 , 1 { ) ( : , of the edges of G, where adjacent edges may be colored the same. A path in G is called a rainbow path if no two edges of it are colored the same. G is rainbow connected if G contains a rainbow v u − path for every two vertices u and v in it. The minimum k for which there ...

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...

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 edge-connectivity matrix of a weighted graph is the whose off-diagonal v-w entry weight minimum edge cut separating vertices v and w. Its computation classical topic combinatorial optimization since at least seminal work Gomory Hu. In this article, we investigate spectral properties these matrices. particular, provide tight bounds on smallest eigenvalue energy. Moreover, study eigenvector s...