Integral Sum Graphs from a Class of Trees

Abstract

A graph \(G = (V, E)\) is said to be an \(integral \;sum \;graph\) ( respectively, \(sum \;graph\)) if there is a labeling \(f\) of its vertices with distinct integers ( respectively, positive integers) , so that for any two vertices \(u\) and \(v\), \(uv\) is an edge of \(G\) if and only if \(f(u) + f(v) = f(w)\) for some other vertex \(w\). For a given graph \(G\), the \(integral\; sum\; number\) \(\zeta = \zeta(G)\) (respectively, \(sum\; number\) \(\sigma = \sigma(G)\) ) is defined to be the smallest number of isolated vertices which when added to \(G\) result in an integral sum graph (respectively, sum graph). In a graph \(G\), a vertex \(v \in V(G)\) is said to a \(hanging\; vertex\) if the degree of it \(d(v) = 1\). A path \(P \subseteq G\), \(P = x_ox_1x_2\ldots x_t\), is said to be a \(hanging\; path\) if its two end vertices are respectively a hanging vertex \(x_o\) and a vertex \(x_t\) whose degree \(d(x_t) \neq 2\) where \(d(x_j) = 2 (j = 1,2,\ldots,t – 1)\) for every other vertex of \(P\). A hanging path \(P\) is said to be a tail of \(G\), denoted by \(t(G)\), if its length \(|t(G)|\) is a maximum among all hanging paths of \(G\). In this paper, we prove \(\zeta(T_3) = 0\), where \(T_3\) is any tree with \(|t(T_3)| \geq 3\). The result improves a previous result for integral sum trees from identification of Chen\((1998)\).