On infinite families of $[P_n]$-irregular graphs

Source abstract

This paper presents the first systematic study of $[F]$-irregular graphs, a concept that parallels classical $F$-irregularity. For a fixed graph $F$, a graph $G$ is $[F]$-irregular if the numbers of its induced subgraphs isomorphic to $F$ containing a given vertex are pairwise distinct for all vertices of $G$. We prove that there exist infinitely many $[P_n]$-irregular graphs for any path $P_n$ of order $n \ge 3$. We establish that a non-trivial $[P_3]$-irregular graph of order $k$ exists if and only if $k \ge 7$. Finally, we propose the Strong Conjecture on $[F]$-irregular graphs.