It has been shown by Voiculescu and Biane that the analytic subordination property holds for free additive and multiplicative convolutions. In this paper, we present an operatorial approach to subordination for free multiplicative convolutions. This study is based on the concepts of ‘freeness with subordination’, or ‘s-free independence’, and ‘orthogonal independence’, introduced recently in th...

An (α, β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v,H) ≤ α ·dist(u, v,G)+β for every pair (u, v) ∈ V × V , where dist(u, v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1, 0)-spanner of size O(n). This size-stretch bound is essentially optimal by the g...

An (α, β)-spanner of an n-vertex graph G = (V,E) is a subgraph H of G satisfying that dist(u, v,H) ≤ α ·dist(u, v,G)+β for every pair (u, v) ∈ V × V , where dist(u, v,G′) denotes the distance between u and v in G′ ⊆ G. It is known that for every integer k ≥ 1, every graph G has a polynomially constructible (2k − 1, 0)-spanner of size O(n). This size-stretch bound is essentially optimal by the g...

Let G be a finite multiplicative group with identity e and ( ) N G be the Neutrosophic group with indeterminate I . We denote by   , Ne G I , the Neutrosophic graph of G , ( ) N G and I . In this paper, we study the graph   , Ne G I and its properties. Among the results, it is shown that for any finite multiplicative group G ,   , Ne G I is a connected graph of diameter less than or equal...

In the year 1971, W. G. Lister [14] introduced the notion of ternary ring. Later in the year 2014, J. R. Castillo and Jocelyn S. ParaderoVilela [2] introduced the notion of krasner ternary hyperring. In [6] we introduce the notion of multiplicative ternary hyperring. In this paper we study prime hyperideals in a multiplicative ternary hyperring. We obtain some Characterizations of prime hyperid...

A graph K is called multiplicative if whenever a categorical product of two graphs admits a homomorphism to K, then one of the factors also admits a homomorphism to K. We prove that all circular graphs Kk/d such that k/d < 4 are multiplicative. This is done using semi-lattice endomorphism in (the skeleton of) the category of graphs to prove the multiplicativity of some graphs using the known mu...

Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

Abstract Analogues to multiplicative Zagreb indices in this paper two new type of eccentricity related topological index are introduced called the first and second multiplicative Zagreb eccentricity indices and is defined as product of squares of the eccentricities of the vertices and product of product of the eccentricities of the adjacent vertices. In this paper we give some upper and lower b...

In this paper, we are giving MATLAB program to find the energy and Wiener index of unit graphs. Our program demonstrates an intrinsic relationship between the elements of ring and structural properties of graphs. The unit graph G(Zn) turns out to be strongly regular when n = 2k. Several new directions for further research are also indicated by means of raising problems.

It is proved that every strongly connected directed graph with n nodes and at least ⌊(n + 1)/4⌋ edges must contain an even cycle. This is best possible, and the structure of extremal graphs is discussed.