Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21285220201201On strongly 2-multiplicative graphs1791901402810.22049/cco.2020.26647.1127END.D.SomashekaraDepartment of Studies in Mathematics, University of Mysore
Manasagangotri, Mysore-570006, IndiaH.E.RaviDepartment of Studies in Mathematics,
University of Mysore, Manasagangotri, Mysore-570006C.R.VeenaDepartment of Mathematics, JSS College of Arts, Commerce and Science,
Mysore-570025, India0000-0001-9804-659xJournal Article20190822A simple connected graph $G$ of order $nge 3$ 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. http://comb-opt.azaruniv.ac.ir/article_14028_5ef7f3d3936254933ebe84c316170400.pdf