A fundamental theorem of Wilson states that, for every graph F , every sufficiently large F -divisible clique has an F -decomposition. Here a graph G is F -divisible if e(F ) divides e(G) and the greatest common divisor of the degrees of F divides the greatest common divisor of the degrees of G, and G has an F -decomposition if the edges of G can be covered by edge-disjoint copies of F . We ext...

A topological index is a numeric quantity associated with chemical structure that attempts to link the various physicochemical properties, reactivity, or biological activity. Let R be commutative ring identity, and Z*(R) set of all non-zero zero divisors R. Then, Γ(R) said zero-divisor graph if only a·b=0, where a,b∈V(Γ(R))=Z*(R) (a,b)∈E(Γ(R)). We define a∼b a·b=0 a=b. ∼ always reflexive symmet...

Let G = (V, E) be a graph with p vertices and q edges. A has product-set labeling if there exist an injective function f: V(G) → P (N) such that the induced edge f∗: P(N) is defined as f∗ (μν) f (μ) ∗ (ν) ∀μ, ν ∈ E(G).f {ab: (μ), b }. In this paper we investigate how works for zero-divisor line of graphs.

Let G = Γ(S) be a semigroup graph, i.e., zero-divisor graph of S with zero element 0. For any adjacent vertices x, y in G, denote C(x,y) {z∈V(G) | N (z) {x,y}}. Assume that there exist two y, vertex s∈C(x,y) and z such d (s,z) 3. This paper studies algebraic properties graphs Γ(S), giving some sub-semigroups ideals S. It constructs classes classifies all the property cases.

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.