Following Harary, the conditional connectivity (edge-connectivity) of a graph with respect to a given graph-theoretic property is the minimum cardinality of a set of vertices (edges), if any, whose deletion disconnects the graph and every remaining component has such a property. We study the case in which all these components are different from a tree whose order is not greater than n. For inst...

The k-th semi total point graph of a graph G, , ‎is a graph‎ obtained from G by adding k vertices corresponding to each edge and‎ connecting them to the endpoints of edge considered‎. ‎In this paper‎, a formula for Laplacian polynomial of in terms of‎ characteristic and Laplacian polynomials of G is computed‎, ‎where is a connected regular graph‎.The Kirchhoff index of is also computed‎.

For designing reliable and efficient communications networks, the problem of constructing a maximally connected d-regular digraph (directed graph) with a small diameter is investigated. A maximally connected d-regular digraph with a diameter at most two larger than the lower bound for any number of nodes n > 2d and any d > 3 is constructed. Since the diameter of this digraph is quasiminimal (at...

We address two practical issues, namely smoothing factor selection and efficient implementation of thresholding with hysteresis, in implementing Canny’s edge detector. The smoothing factor of the Gaussian kernel should be chosen to maximize the discrete version of Canny’s original criteria. Thresholding with hysteresis should be implemented using an efficient connected component analysis algori...

We show that for every ℓ , there exists d such 3-edge-connected graph with minimum degree can be edge-partitioned into paths of length (provided its number edges is divisible by ). This improves a result asserting 24-edge-connectivity and high provides partition. best possible as 3-edge-connectivity cannot replaced 2-edge connectivity.

In this paper, we determine all of connected normal edge-transitive Cayley graphs on non-abelian groups with order $4p^2$, where $p$ is a prime number.

Let G be a finite connected graph. The degree distance D′(G) of G is defined as ∑ {u,v}⊆V (G)(deg u + deg v) dG(u, v), where degw is the degree of vertex w and dG(u, v) denotes the distance between u and v in G. In this paper, we give asymptotically sharp upper bounds on the degree distance in terms of order and edge-connectivity.

The power graph of a group G is the graph whose vertex set is G and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian p-groups, dihedral groups and dicyclic groups are obtained. It is ascertained that the edge-connectivity and minimum degree of power graphs are equal, and consequently...

A. Frank (Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5(1), 22–53, 1992) developed a method to solve edgeconnectivity augmentation problems. His paper has stimulated further research in a number of directions, including many interesting generalizations. This paper surveys the current State of the Art on the edge-connectivity augmentation problem. Recent exte...