Some results on the spum and the integral spum of graphs

A finite simple graph G is called a sum graph (integral sum graph) if there is a bijection f from the vertices of G to a set of positive integers S (a set of integers S) such that uv is an edge of G if and only if f(u)+f(v)∈S. For a connected graph G, the sum number (the integral sum number) of G, denoted by σ(G) (ζ(G)), is the minimum number of isolated vertices that must be added to G so that the resulting graph is a sum graph (an integral sum graph). The spum (the integral spum) of a graph G is the minimum difference between the largest and smallest integer in any set S that corresponds to a sum graph (integral sum graph) containing G. We investigate the spum and integral spum of several classes of graphs, including complete graphs, symmetric complete bipartite graphs, star graphs, cycles, and paths. We also give sharp lower bounds for the spum and the integral spum of connected graphs.