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 called $F$-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 $|F|\geqslant 3$, there exist infinitely many $F$-irregular graphs. In the present 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 $3$.