Hub Dominating Sets and Hub Domination Polynomials of the Lollipop Graph L_(n,1)

Abstract

Let G=(V,E)be a simple graph. Let  HD(G,j) be the family of hub dominating sets in G with cardinality j. Then, the polynomial,

HD(G,x)=∑_(j=hd(G))^|V(G)|▒〖hd(G,j)x^j 〗

is called the  hub domination polynomial of G  where  hd(G,j) is the number of  hub dominating sets of  G of cardinality j and  hd(G) is the hub domination number of  G . Let  L_(n,1) denotes the Lollipop graph with  n+1 vertices and  HD(L_(n,1),j) denotes the family of hub dominating sets of L_(n,1)with cardinality j. Then, the polynomial,

HD(L_(n,1),x)=∑_(j=hd(L_(n,1) ))^|V(L_(n,1) ) |▒〖hd(L_(n,1),j) x^j 〗

is called the hub domination polynomial of L_(n,1)where  hd(L_(n,1),j) is the number of  hub dominating sets of L_(n,1)of cardinality j and  hd(L_(n,1) )is hub domination number of  L_(n,1).In this paper, we obtain a recursive formula for hd(L_(n,1),j). Using this recursive formula, we construct the hub domination polynomial ofL_(n,1) as,

 HD(L_(n,1),x)=∑_(j=1)^(n+1)▒〖hd(L_(n,1),j)x^j 〗

where  hd(L_(n,1),j) is the number of  hub dominating sets of L_(n,1)of cardinality j and some of the properties of this polynomial also have been studied.