Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 5 2 2020 12 01 On strongly 2-multiplicative graphs 179 190 14028 10.22049/cco.2020.26647.1127 EN D.D. Somashekara Department of Studies in Mathematics, University of Mysore Manasagangotri, Mysore-570006, India H.E. Ravi Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore-570006 C.R. Veena Department of Mathematics, JSS College of Arts, Commerce and Science, Mysore-570025, India Journal Article 2019 08 22 A simple connected graph $G$ of order $n\ge 3$ is a strongly 2-multiplicative if there is an injective mapping $f:V(G)\rightarrow \{1,2,\ldots,n\}$ such that the induced mapping $h:\mathcal{A} \rightarrow \mathbb{Z}^+$ defined by $h(\mathcal{P})= \prod_{i=1}^{3} f({v_j}_i)$, where $j_1,j_2,j_{3}\in \{1,2,\ldots,n\}$, and $\mathcal{P}$ is the path homotopy class of paths having the vertex set $\{ v_{j_1}, v_{j_2},v_{j_{3}} \}$, is injective. Let $\Lambda(n)$ be the number of distinct path homotopy classes in a strongly 2-multiplicative graph of order $n$. In this paper we obtain an upper bound and also a lower bound for $\Lambda(n)$. Also we prove that triangular ladder, $P_{2} \bigodot C_{n}$, $P_{m}\bigodot P_{n}$, the graph obtained by duplication of an arbitrary edge by a new vertex in path $P_{n}$ and the graph obtained by duplicating all vertices by new edges in a path $P_{n}$ are strongly 2-multiplicative.  https://comb-opt.azaruniv.ac.ir/article_14028_5ef7f3d3936254933ebe84c316170400.pdf