On the Uphill Domination Polynomial of Graphs

A path [ ] 1 2 , , , k v v v π=  in a graph ( ) , G V E = is an uphill path if ( ) ( ) 1 i i deg v deg v + ≤ for every 1 i k ≤ ≤ . A subset ( ) S V G ⊆ is an uphill dominating set if every vertex ( ) i v V G ∈ lies on an uphill path originating from some vertex in S. The uphill domination number of G is denoted by ( ) up G γ and is the minimum cardinality of the uphill dominating set of G. In this paper, we introduce the uphill domination polynomial of a graph G. The uphill domination polynomial of a graph G of n vertices is the polynomial ( ) ( ) ( ) , , up n i i G UP G x up G i x γ= = ∑ , where ( ) , up G i is the number of uphill dominating sets of size i in G, and ( ) up G γ is the uphill domination number of G, we compute the uphill domination polynomial and its roots for some families of standard graphs. Also, ( ) , UP G x for some graph operations is obtained.

is the uphill domination number of G, we compute the uphill domination polynomial and its roots for some families of standard graphs. Also,

Introduction
In this paper, we are concerned with simple graphs which are finite, undirected with no loops nor multiple edges. Throughout this paper, we let  with , , 0 s t k ≥ and 2 2 1 n s t k = + + + vertices is defined by consisting of s triangles, t pendent paths of length 2 and k pendent edges, sharing a common vertex. Any terminology not mentioed here we refer the reader to [2].
A set S V ⊆ of vertices in a graph G is called a dominating set if every vertex v V ∈ is either v S ∈ or v is adjacent to an element of S, The uphill dominating set "UDS" is a set S V ⊆ having the property that every vertex v V ∈ lies on an uphill path originating from some vertex in S. The uphill domination number of a graph G is denoted by and is defined to be the minimum cardinality of the UDS of G. Moreover, it's customary to denote the UDS having the minimum cardinality by ( ) up G γ -set, for more details in domination see [3] and [4].
Representing a graph by using a polynomial is one of the algebraic representations of a graph to study some of algebraic properties and graph's structure. In general graph polynomials are a well-developed area which is very useful for analyzing properties of the graphs.
The domination polynomial [5] and the uphill domination of a graph [6], motivated us to introduce and study the uphill domination polynomial and the uphill domination roots of a graph.

Uphill Domination Polynomial
Definition 2.1. For any graph G of n vertices, the uphill domination polynomial of G is defined by Since the first coefficient of the polynomial is n, then it is easily verified that for every Corollary 2.4. Let G ba a graph with s vertices. If G is a cycle s C or complete graph s K , then Corollary 2.5. The uphill domination polynomial for the regular graph with sk vertices is given by Proof. By using mathematical induction we found that for 1 m = the statement is true and the proof is trivial. Suppose that the statement is true when Now, we prove that the statement is true when 1 m k = + . Let G consists of 1 k + components that mean , , , k r r r +  represent the uphill domination number for the components of G respectively, such that ( ) , up G r is exactly equal the number of way for choosing an UDS of size 1 r in 1 G and an UDS of size 2 r in 2 G and so on. Hence, In the same argument we can proof for all Thus, for 1 m k = + the statement is true and the proof is done.
Theorem 2.8. For any path n P with 3 n ≥ vertices, Proof. Let G be a path graph n P with 3 n ≥ . We know that Proof. Let G be a bistar graph The generalization of Theorem 0.12 is the following result.
Theorem 2.14. For any graph          Figure 2. A Book Graph Bm.

Uphill Domination Polynomials of Graphs under Some Binary Operations
And so on, we use the same argument until ( )