Some Parameters on Neighborhood Number of A Graph

A set S⊆V is a neighborhood set of G , if G=⋃v∈S〈N[v]〉, where 〈N[v]〉 is the sub graph of G induced by v and all vertices adjacent to v . The neighborhood number η(G) of G is the minimum cardinality of a neighborhood set of G. In this paper, we extended the concept of neighborhood number and its relationship with other related parameters are explored

Affirmative domination in graphs.

A function f : V → {−1, 0, 1} is an affirmative dominating function of graph G satisfying the conditions that for every vertex u such that f(u) = 0 is adjacent to at least one vertex v for which f(v) = 1 and P u∈N(v) f(u) ≤ 1 for every v ∈ V . The affirmative domination number γa(G) =max{w(f) : f is affirmative dominating function}. In this paper, we initiate the study of affirmative and strongly affirmative dominating functions. Here, we obtain some properties of these new parameters and also determine exact values of some special classes of graph

Bounds on perfect k-domination in trees: An algorithmic approach

Let k be a positive integer and G = (V,E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G if every vertex v of G not in D is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γkp(G). In this paper, a sharp bound for γ kp(T) is obtained where T is a tree

Roman Domination in Complementary Prism Graphs

A Roman domination function on a complementary prism graph GGc is a function f : V [ V c ! {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number R(GGc) of a graph G = (V,E) is the minimum of Px2V [V c f(x) over such functions, where the complementary prism GGc of G is graph obtained from disjoint union of G and its complement Gc by adding edges of a perfect matching between corresponding vertices of G and Gc. In this paper, we have investigated few properties of R(GGc) and its relation with other parameters are obtaine

On coefficients of edge domination polynomial of a graph

An edge domination polynomial of a graph G is the polynomial where de(G, t) is the number of edge dominating sets of G of cardinality t. In this paper, we provide tables which contain coefficient of edge domination polynomial of path and cycle. Also, certain properties of edge dominating polynomial are given

Complementary total domination in graphs

Let D be a minimum total dominating set of G. If V−D contains a total dominating set (TDS) say S of G, then S is called a complementary total dominating set with respect to D. The complementary total domination number γ ct (G) of G is the minimum number of vertices in a complementary total dominating set (CTDS) of G .In this paper, exact values of γ ct (G) for some standard graphs are obtained. Also its relationship with other domination related parameters are investigated

The Dual Neighborhood Number of a Graph

A set S ⊆ V (G) is a neighborhood set of a graph G = (V,E), if G = v∈SN[v]�, where N[v]� is the sub graph of a graph G induced by v and all vertices adjacent to v. The dual neighborhood number η+2(G) = Min. {|S1|+ |S2| : S1, S2 are two disjoint neighborhood set of G}. In this paper, we extended the concept of neighborhood number to dual neighborhood number and its relationship with other neighborhood related parameters are explored.

Multiplicative Product Connectivity and Sum Connectivity Indices of Chemical Structures in Drugs

In Chemical sciences, the multiplicative connectivity indices are used in the analysis of drug molecular structures which are helpful for chemical and medical scientists to find out the chemical and biological characteristics of drugs. In this paper, we compute the multiplicative product and sum connectivity indices of some important nanostar dendrimers which appeared in nanoscience

On the roots of total domination polynomial of graphs, II

Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. A root of $D_t(G, x)$ is called a total domination root of $G$. The set of total domination roots of graph $G$ is denoted by $Z(D_t(G,x))$. In this paper we show that $D_t(G,x)$ has $\delta-2$ non-real roots and if all roots of $D_t(G,x)$ are real then $\delta\leq 2$, where $\delta$ is the minimum degree of vertices of $G$. Also we show that if $\delta\geq 3$ and $D_t(G,x)$ has exactly three distinct roots, then $Z(D_t(G,x))\subseteq \{0, -2\pm \sqrt{2}i, \frac{-3\pm \sqrt{3}i}{2}\}$. Finally we study the location roots of total domination polynomial of some families of graphs.Comment: 10 pages, 5 figure

PBIB-designs and association schemes arising from minimum bi-connected dominating sets of some special classes of graphs

A dominating set D of a connected graph G=(V,E) is said to be bi-connected dominating set if the induced subgraphs of both ⟨D⟩ and ⟨V−D⟩ are connected. The bi-connected domination number γbc(G) is the minimum cardinality of a bi-connected dominating set. A γbc -set is a minimum bi-connected dominating set of G. In this paper, we obtain the Partially Balanced Incomplete Block (PBIB)-designs with m = 1, 2, 3, 4 and ⌊p2⌋ association schemes arising from γbc -sets of some special classes of graphs