Restricted connectivity of cartesian products of some graphs

A vertex cut S of G is said to be a restricted vertex cut if NG(u) - S , ? for any vertex u of G. If G has a restricted vertex cut, then the minimum cardinality of them called the restricted connectivity of G and we denote the restricted connectivity of G by ′(G); this is a more refined index than the connectivity parameter (G). In this paper, we prove that 2 1 + 2 2 ?? 2 ?? t1 ?? t2 ′(G1 G2) Δ1 + 1 + 2 2 ?? 2, where, for i = 1; 2, Gi is a maximally connected (Kti+1 + K2) - free graph such that (Gi) 2, V(Gi) 2 i ?? ti, and Δ1 + 2 Δ2 + 1,where i = i(G) and Δi = Δi(G) for i = 1; 2.電気通信大学201

ON THE LAPLACIAN SPECTRA OF PRODUCT GRAPHS

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs

On linkedness in the Cartesian product of graphs

We study linkedness of the Cartesian product of graphs and prove that the product of an a-linked and a b-linked graph is (a + b - 1)-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of products of paths and products of cycles

On subgraphs of Cartesian product graphs and S-primeness

AbstractIn this paper we consider S-prime graphs, that is the graphs that cannot be represented as nontrivial subgraphs of nontrivial Cartesian products of graphs. Lamprey and Barnes characterized S-prime graphs via so-called basic S-prime graphs that form a subclass of all S-prime graphs. However, the structure of basic S-prime graphs was not known very well. In this paper we prove several characterizations of basic S-prime graphs. In particular, the structural characterization of basic S-prime graphs of connectivity 2 enables us to present several infinite families of basic S-prime graphs. Furthermore, simple S-prime graphs are introduced that form a relatively small subclass of basic S-prime graphs, and it is shown that every basic S-prime graph can be obtained from a simple S-prime graph by a sequence of certain transformations called extensions

Group connectivity of semistrong product of graphs

Abstract For a 2-edge-connected graph G, the group connectivity number of G is defined as Λ g (G) = min{k : G is A-connected for every abelian group with |A| ≥ k}. Let G • H denote the semistrong product of two graphs G and H. In this paper, we extend the result of Yan et al. [Int. J. Algebra 4 (2010), 1185-1200] on group connectivity in Cartesian product of graphs to semistrong products and make a slightly stronger conclusion

On Linkedness of Cartesian Product of Graphs

We study linkedness of Cartesian product of graphs and prove that the product of an $a$-linked and a $b$-linked graphs is $(a+b-1)$-linked if the graphs are sufficiently large. Further bounds in terms of connectivity are shown. We determine linkedness of product of paths and product of cycles

On the super connectivity of Kronecker products of graphs

In this paper we present the super connectivity of Kronecker product of a general graph and a complete graph.Comment: 8 page