Let C be a C4-design of order n and index λ, on the vertex set V , |V | = n. If V1∪· · ·∪Vm = V is a partition of the vertex set, such that the intersections of the C ∈ C with Vi form a P3-design of order |Vi| and the same index λ, for each 1 ≤ i ≤ m, then 2 ≤ m ≤ log3(2n+1). The minimum bound is best possible for every λ. The maximum bound is best possible for λ = 2, and hence also for every e...

Let G be a connected graph with vertex set V (G) and edge set E(G). The eccentric connectivity index of G, denoted by ξc(G), is defined as ∑ v∈V (G) deg(v)ec(v), where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. In this paper, we propose the edge version of the above index, the edge eccentric connectivity index of G, denoted by ξc e(G), which is defined as ξc e(G) = ∑ f∈E(...

Let G = (V,E) be a graph, where V(G) is a non-empty set of vertices and E(G) is a set of edges, e = uv∈E(G), d(u) is degree of vertex u. Then the first Zagreb polynomial and the first Zagreb index Zg1(G,x) and Zg1(G) of the graph G are defined as ( ) ( ) ∑ u v u d v E G d x + ∈ and ( ) ( ) ∑ e u G v uv E d d = ∈ + respectively. Recently Ghorbani and Hosseinzadeh introduced the first Eccentric Z...

for every positive integer k, a set s of vertices in a graph g = (v;e) is a k- tuple dominating set of g if every vertex of v -s is adjacent to at least k vertices and every vertex of s is adjacent to at least k - 1 vertices in s. the minimum cardinality of a k-tuple dominating set of g is the k-tuple domination number of g. when k = 1, a k-tuple domination number is the well-studied domination...

Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. The Wiener Polarity index ...

a lot of research and various techniques have been devoted for finding the topologicaldescriptor wiener index, but most of them deal with only particular cases. there exist threeregular plane tessellations, composed of the same kind of regular polygons namely triangular,square, and hexagonal. using edge congestion-sum problem, we devise a method to computethe wiener index and demonstrate this m...

A 2-rainbow dominating function ( ) of a graph  is a function  from the vertex set  to the set of all subsets of the set  such that for any vertex  with  the condition  is fulfilled, where  is the open neighborhood of . A maximal 2-rainbow dominating function on a graph  is a 2-rainbow dominating function  such that the set is not a dominating set of . The weight of a maximal    is the value . ...