In the definition of the graph parameters μ(G) and ν(G), introduced by Colin de Verdière, and in the definition of the graph parameter ξ(G), introduced by Barioli, Fallat, and Hogben, a transversality condition is used, called the Strong Arnol’d Hypothesis. In this paper, we define the Strong Arnol’d Hypothesis for linear subspaces L ⊆ R with respect to a graph G = (V,E), with V = {1, 2, . . . ...

This paper introduces a fascinating maze based solely on the distribution of the prime numbers. Although it was originally designed as a simple puzzle, the maze revealed some rather startling properties of the primes. The rules are so simple and natural that traversing the maze seems more like exploring a natural cave formation than a maze of human design. We will describe this maze using the l...

Let R be a commutative ring and let ? Z n the zero divisor graph of id="M2"> R , whose vertices are nonzero divisors id="M3"> such that two id="M4"> u , v adjacen...

Graph theory is a branch of algebra that growing rapidly both in concept and application studies. This graph can be used chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There currently bridge between graphs algebra, especially an algebraic structures, namely algebra. One researchs on formed by prime ring elements or called over R. The commuta...

7 A homogeneous set of a graph G is a set X of vertices such that 2 ≤ |X| < |V (G)| 8 and no vertex in V (G) − X has both a neighbor and a non-neighbor in X. A graph 9 is prime if it has no homogeneous set. We present an algorithm to decide whether a 10 class of graphs given by a finite set of forbidden induced subgraphs contains infinitely 11 many non-isomorphic prime graphs. 12

Characterizing regular covers of edge-transitive or arc-transitive graphs is currently a hot topic in algebraic graph theory. In this paper, we will classify arctransitive regular cyclic covers of the complete bipartite graph Kp,p for each odd prime p. The classification consists of four infinite families of graphs. In particular, such covers exist for each odd prime p. The regular elementary a...