In this paper, the authors investigate the algebraic connectivity of connected graphs, and determine the graph which has the minimum algebraic connectivity among all connected graphs of order n with given stability number α ≥ d 2 e, or covering number, respectively.

This paper is a survey of the second smallest eigenvalue of the Laplacian of a graph G, best-known as the algebraic connectivity of G, denoted a(G). Emphasis is given on classifications of bounds to algebraic connectivity as a function of other graph invariants, as well as the applications of Fiedler vectors (eigenvectors related to a(G)) on trees, on hard problems in graphs and also on the com...

The generalized k-connectivity κk(G) of a graphG, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let G and H be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized 3-connectivity of the Cartesian product graph G H and proposed a conjecture for the case that H is 3-connected. In this paper, we give two...

For a connected graph G = (V (G), E(G)), a vertex set S ⊆ V (G) is a k-restricted vertex-cut if G − S is disconnected such that every component of G − S has at least k vertices. The k-restricted connectivity κk(G) of the graph G is the cardinality of a minimum k-restricted vertex-cut of G. In this paper, we give the 3-restricted connectivity and the 4-restricted connectivity of the Cartesian pr...

Motivated by challenges related to domination, connectivity, and information propagation in social and other networks, we initiate the study of the Vector Connectivity problem. This problem takes as input a graph G and an integer kv for every vertex v of G, and the objective is to find a vertex subset S of minimum cardinality such that every vertex v either belongs to S, or is connected to at l...

By a result of McKenzie [4] finite directed graphs that satisfy certain connectivity and thinness conditions have the unique prime factorization property with respect to the cardinal product. We show that this property still holds under weaker connectivity and stronger thinness conditions. Furthermore, for such graphs the factorization can be determined in polynomial time.

The Immerman-Szelepcsenyi Theorem uses an algorithm for co-stconnectivity based on inductive counting to prove that NLOGSPACE is closed under complementation. We want to investigate whether counting is necessary for this theorem to hold. Concretely, we show that Nondeterministic Jumping Graph Autmata (ND-JAGs) (pebble automata on graphs), on several families of Cayley graphs, are equal in power...

The generalized k-connectivity of a graph G, denoted by κk(G), is the minimum number internally edge disjoint S-trees for any S⊆V(G) and |S|=k. natural extension classical connectivity plays key role in applications related to modern interconnection networks. burnt pancake BPn godan EAn are two kinds Cayley graphs which posses many desirable properties. In this paper, we investigate 3-connectiv...

We will present a new method, which enables us to find threshold functions for many properties in random intersection graphs. This method will be used to establish sharp threshold functions in random intersection graphs for k–connectivity, perfect matching containment and Hamilton cycle containment. keywords: random intersection graph, threshold functions, connectivity, Hamilton cycle, perfect ...