Proof of the Strong Conjecture about $$F$$-Irregular Graphs in the Class of Graphs $$\{F\}$$ of Diameter 2

Abstract

Let $$F$$ and $$G$$ be simple finite undirected graphs. A graph $$G$$ is said to be $$F$$ -irregular if any two of its distinct vertices belong to different numbers of copies of $$F$$ in $$G$$ . According to the strong conjecture about $$F$$ -irregular graphs (Dovzhenok, Filuta, Chuhai), for any connected graph $$F$$ of order at least three, there exist infinitely many $$F$$ -irregular graphs. In this paper, the strong conjecture about $$F$$ -irregular graphs is confirmed in the class of graphs $$\{F\}$$ of diameter $$2$$ . It is proved that, for every graph $$F$$ of diameter $$2$$ , there exists an infinite series of $$F$$ -irregular graphs of diameter $$3$$ .