In this paper we introduce and study a type of Cayley graph – subnormal graph. We prove that 2-arc transitive is normal or cover complete bipartite $\mathbf{K}_{p^d,p^d}$ with $p$ prime. Then obtain generic method for constructing half-symmetric (namely edge but not arc transitive) graphs.

Let $D$ be a weighted oriented graph and $I(D)$ its edge ideal. If contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary sufficient for the equality ordinary symbolic powers ideal having each in some it. characterize naturally unicyclic graphs with unique cycles even their ideals. D^{\prime} obtained from afte...

In this paper a bijection between the set of prime ideals of a Leavitt path algebra L K (E) and a certain set which involves maximal tails in E and the prime spectrum of K[x, x −1 ] is established. Necessary and sufficient conditions on the graph E so that the Leavitt path algebra L K (E) is primitive are also found.

We introduce compactness theorems for generalized colorings and derive several particular compactness theorems from them. It is proved that the theorems and many of their consequences are equivalent in ZF set theory to BPI, the Prime Ideal Theorem for Boolean algebras. keywords: generalized graph colorings, compactness, prime ideal theorem MSC: 05C15; 03E25 .

A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are ba...

A graph G(V,E) with vertex set V is said to have a prime labeling if its vertices are labeled with distinct integers 1, 2, . . . , |V | such that for each edge xy ∈ E the labels assigned to x and y are relatively prime. A prime cordial labeling of a graph G with vertex set V is a bijection f from V to {1, 2, . . . , |V |} such that if each edge uv is assigned the label 1 if gcd(f(u), f(v)) = 1 ...

Let G be a finite group, and let Δ(G) the prime graph built on its set of conjugacy class sizes: this is (simple undirected) whose vertices are numbers dividing some size G, two distinct p, q adjacent if only pq divides G. In paper, we characterize structure those groups block square.

We study the concept of stabilization with internal loop for infinite-dimensional discrete timevarying systems in the framework of nest algebra. We originally give a parametrization of all stabilizing controllers with internal loop, and it covers the parametrization of canonical or dual canonical controllers with internal loop obtained before. We show that, in practical application, the control...

In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomialtime algorithm, Algorithmica 78(1):342–377, 2017], we presented a characterization of the linear rank-width of distance-hereditary graphs, from which we derived an algorithm to compute it in polynomial time. In this paper, we investigate structural properties of distance-hereditary graphs based on this charac...