Let G = (V (G), E(G)) be a graph with the set of vertices V (G) and the set of edges E(G). A subset S of E(G) is called a k-nearly independent edge subset if there are exactly k pairs of elements of S that share a common end. Zk(G) is the number of such subsets. This paper studies Z1. Various properties of Z1 are discussed. We characterize the two n-vertex trees with the smallest Z1, as well as the one with the largest value. A conjecture on the n-vertex tree with the second-largest Z1 is proposed.

A visual cryptography scheme based on a given graph G is a method to distribute a secret image among the vertices of G, the participants, so that a subset of participants can recover the secret image if they contain an edge of G, by stacking their shares, otherwise they can obtain no information regarding the secret image. In this paper a Maximal independent sets of the graph G was applied to propose a lower bound on the pixel expansion of visual cryptography schemes with graph access structure G (G). In addition a lower bound on the pixel expansion of basis matrices C5 and Peterson graph access structure were presented.

In this article, we extend the concept of divisors to ideals of Noetherian rings, more generally, to submodules of finitely generated modules over Noetherian rings. For a submodule $N$ of a finitely generated module $M$ over a Noetherian ring, we say a submodule $K$ of $M$ is a regular divisor of $N$ in $M$ if $K$ occurs in a regular prime extension filtration of $M$ over $N$. We show that a submodule $N$ of $M$ has only a finite number of regular divisors in $M$. We also show that an ideal $\mathfrak b$ is a regular divisor of a non-zero ideal $\mathfrak a$ in a Dedekind domain $R$ if and only if $\mathfrak b$ contains $\mathfrak a$. We characterize regular divisors using some ordered sequences of prime ideals and study their various properties. Lastly, we formulate a method to compute the number of regular divisors of a submodule by solving a combinatorics problem.

We investigated Maximal Prym varieties on  nite  elds by attaining their upper bounds on the number of rational points. This concept gave us a motivation for de ning a generalized de ni-tion of Maximal curves i. e., Maximal morphisms. By MAGMA, we give some non-trivial examples of Maximal morphisms that results in non-trivial examples of Maximal Prym varieties.

This article explains a particular category of neutrosophic subsets of G-submodules, specifically r-neutrosophic subsets, where r ∈ [0, 1]. The algebra of r-neutrosophic subsets of G-submodules is dis-cussed, along with some fundamental characteristics of their sum. Definitions and theorems related to this concept are provided to clarify the properties of an arbitrary non-empty family of r-neutrosophic subsets of G-submodules.

This article explains a particular category of neutrosophic subsets of G-submodules, specifically r-neutrosophic subsets, where r ∈ [0, 1]. The algebra of r-neutrosophic subsets of G-submodules is dis-cussed, along with some fundamental characteristics of their sum. Definitions and theorems related to this concept are provided to clarify the properties of an arbitrary non-empty family of r-neutrosophic subsets of G-submodules.