The (mod, integral) sum numbers of fans and [formula omitted

Abstract

Let N ( Z ) denote the set of all positive integers (integers). The sum graph G + ( S ) of a finite subset S ⊂ N ( Z ) is the graph ( S , E ) with uv ∈ E if and only if u + v ∈ S . A graph G is said to be an (integral) sum graph if it is isomorphic to the sum graph of some S ⊂ N ( Z ) . The (integral) sum number σ ( G ) ( ζ ( G ) ) of G is the smallest number of isolated vertices which when added to G result in an (integral) sum graph. A mod sum graph is a sum graph with S ⊂ Z m ⧹ { 0 } and all arithmetic performed modulo m where m ⩾ | S | + 1 . The mod sum number ρ ( G ) of G is the least number ρ of isolated vertices ρ K 1 such that G ∪ ρ K 1 is a mod sum graph. In this paper, we prove that for n ⩾ 3 , the n spoked fan F n is an integral sum graph, ρ ( F 4 ) = 1 , ρ ( F n ) = 2 for n ≠ 4 , and σ ( F n ) = 2 , n = 4 , 3 , n = 3 or n ⩾ 6 and n even , 4 , n ⩾ 5 and n odd . We also show that for K n , n - E ( nK 2 ) ( n ⩾ 6 ) , ρ = n - 2 , σ = 2 n - 3 and ζ = 2 n - 5 .