In this paper, we show that every ( 3 k − ) -edge-connected graph G , under a certain degree condition, can be edge-decomposed into factors 1 … such for each vertex v ∈ V i | d / < where ≤ . As an application, deduce 6 three 2 and with unless has exactly one z ⁄ ≡ 0 Next, odd- have the same parity is odd positive integer Finally, give sufficient edge-connectivity condition to factor F specified...

Let m and n be positive integers with n − 1 ≤ m ≤ ( n 2 ) and CG(m,n) be the set of all non-isomorphic connected graphs of order n and size m. The vertex-connectivity and the edge-connectivity of a graph G are denoted by κ(G) and λ(G), respectively. We prove that if π ∈ {κ, λ}, then there exist positive integers a and b such that {π(G) : G ∈ CG(m,n)} = {x ∈ Z : a ≤ x ≤ b}. Thus {π(G) : G ∈ CG(m...

The vertex-connectivity and edge-connectivity of the zero-divisor graph associated to a finite commutative ring are studied. It is shown that the edgeconnectivity of ΓR always coincides with the minimum degree. When R is not local, it is shown that the vertex-connectivity also equals the minimum degree, and when R is local, various upper and lower bounds are given for the vertex-connectivity.

Let G be a simple non-complete graph of order n. The r-component edge connectivity of G denoted as λr(G) is the minimum number of edges that must be removed from G in order to obtain a graph with (at least) r connected components. The concept of r-component edge connectivity generalizes that of edge connectivity by taking into account the number of components of the resulting graph. In this pap...

The enumeration of shortest paths in cubic grid is presented here. The cubic grid considers three neighborhods— 6-neighborhood (face connectivity), 18-neighborhood (edge connectivity), and 26-neighborhood (vertex connectivity). The formulation for distance metrics are given here. L1, D18, and L∞ are the three metrics for 6-neighborhood, 18-neighborhood, and 26-neighborhood. The problem is to fi...

The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. These lead to the conjecture that, if G requires at least t colors, then G must have immersed within it Kt, the complete graph on t vertices. Evidence in support of such a proposition is presented. For each fixed value of t, there can be only a f...

The atom-bond connectivity (ABC) index is a degree-based molecular structure descriptor that can be used for modelling thermodynamic properties of organic chemical compounds. Motivated by its applicable potential, a series of investigations have been carried out in the past several years. In this note we first consider graphs with given edge-connectivity that attain the maximum ABC index. In pa...