The transmission TrG(v) of a vertex v connected graph G is the sum distances between and all other vertices in G. stepwise irregular (STI) if |TrG(u) ? TrG(v)| = 1 holds for each edge uv E(G). In this paper, extremal results on STI graphs with respect to size different metric properties are proved. Two families appear cases, balanced complete bipartite odd order so called hatted cycles.

Given a finite relational language L, a hereditary L-property is a class of finite Lstructures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and re...