Our main purpose of this paper is to study the relationship between Multiplicative maps and almost Multiplicative maps between probabilistic normed algebras. We first derive some properties of invertible elements and their relation with Multiplicative maps. Then we show that every complex homomorphism on elements whose probabilistic norm is equal to 1, is bounded. In the following, we give an open problem about the functionally continuous of unital commutative probabilistic Banach algebra. Finally, we prove that every almost Multiplicative map that is not a Multiplicative map is continuous.

In a Multiplicative hyperring, the multiplication is a hyperoperation, while the addition is a binary operation. In this paper, the notion of derivation on Multiplicative hyperrings is introduced and some related properties are investigated.

A simple connected graph G of order n  3 is a strongly 2-Multiplicative if there is an injective mapping f: V (G)! f1; 2; : : :; ng such that the induced mapping h: A! Z + de ned by h(P) = Q3 i=1 f(vj i ), where j1; j2; j3 2 f1; 2; : : :; ng, and P is the path homotopy class of paths having the vertex set fvj1; vj2; vj3 g, is injective. Let  (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  (n). Also we prove that triangular ladder, P2 J Cn, Pm J Pn, the graph obtained by duplication of an arbitrary edge by a new vertex in path Pn and the graph obtained by duplicating all vertices by new edges in a path Pn are strongly 2-Multiplicative.

Let p be prime and , : x 7! xgx, the Discrete Lambert Map. For k ,1, let V = {0, 1, 2, . . ., pk −,1}. The iteration digraph is a directed graph with V as the vertex set and there is a unique directed edge from u to , (u) for each u 2 V. We denote this digraph by G(g, pk), where g 2 (Z/pkZ), . In this piece of work, we investigate the structural properties and find new results modulo higher powers of primes. We show that if g is of order pd, 1 ,d ,k −,1 then G(g, pk) has pk−, d d 2 e loops. If g = tp + 1, 1 ,t ,pk−, 1 −,1 then the digraph contains p k +1 2 cycles. Further, if g has order pk−, 1 then G(g, pk) has p −,1 cycles of length pk−, 1 and the digraph is cyclic. We also propose explicit formulas for the enumeration of components.

‎The first general Multiplicative Zagreb index $P_1^a(G)$ is the product of the degree of each vertex $v$ in $G$‎, ‎raised to the power $a$ and the second general Multiplicative Zagreb index $P_2^a(G)$ is the product of the degree of each vertex $v$ in $G$‎, ‎raised to the power $a$ times the degree of $v$‎, ‎where $a$ is a non-zero real number‎. ‎In this study‎, ‎we present bounds on the general Multiplicative Zagreb indices for trees and unicyclic graphs‎. ‎We also provide bounds for the first general Multiplicative Zagreb index for trees‎. ‎Additionally‎, ‎we identify all the extremal graphs for each bound mentioned as best as possible‎.

This paper deals with the automatic continuity of Multiplicative polynomial operators on a class of topological algebras. Several results are derived in this direction. We also support our results by some examples.