17 research outputs found
Superconnectivity of Networks Modeled by the Strong Product of Graphs
Maximal connectivity and superconnectivity in a network are two important
features of its reliability. In this paper, using graph terminology, we first
give a lower bound for the vertex connectivity of the strong product of two
networks and then we prove that the resulting structure is more reliable
than its generators. Namely, sufficient conditions for a strong product of two
networks to be maximally connected and superconnected are given.Ministerio de EconomÃa y Competitividad MTM2014-60127-
The generalized 3-edge-connectivity of lexicographic product graphs
The generalized -edge-connectivity of a graph is a
generalization of the concept of edge-connectivity. The lexicographic product
of two graphs and , denoted by , is an important graph
product. In this paper, we mainly study the generalized 3-edge-connectivity of
, and get upper and lower bounds of .
Moreover, all bounds are sharp.Comment: 14 page
The Menger number of the strong product of graphs
The xy-Menger number with respect to a given integer â„“, for every two vertices x, y in
a connected graph G, denoted by ζℓ(x, y), is the maximum number of internally disjoint
xy-paths whose lengths are at most â„“ in G. The Menger number of G with respect to
ℓ is defined as ζℓ(G) = min{ζℓ(x, y) : x, y ∈ V(G)}. In this paper we focus on the
Menger number of the strong product G1 G2 of two connected graphs G1 and G2 with
at least three vertices. We show that ζℓ(G1 G2) ≥ ζℓ(G1)ζℓ(G2) and furthermore, that
ζℓ+2(G1 G2) ≥ ζℓ(G1)ζℓ(G2) + ζℓ(G1) + ζℓ(G2) if both G1 and G2 have girth at least
5. These bounds are best possible, and in particular, we prove that the last inequality is
reached when G1 and G2 are maximally connected graphs.Ministerio de Educación y Ciencia MTM2011-28800-C02-02Generalitat de Cataluña 1298 SGR200