Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

‎In this paper we defined the vertex removable cycle in respect of the following‎, ‎if $F$ is a class of graphs(digraphs)‎ ‎satisfying certain property‎, ‎$G in F $‎, ‎the cycle $C$ in $G$ is called vertex removable if $G-V(C)in in F $.‎ ‎The vertex removable cycles of eulerian graphs are studied‎. ‎We also characterize the edge removable cycles of regular‎ ‎graphs(digraphs).‎    

This is a continuation of previous study initiated by one us on nonlocal vertex bialgebras and smash product algebras. In this paper, we notion right $H$-comodule algebra for bialgebra $H$ give construction deformations algebras with structure compatible $H$-module structure. We also $\phi$-coordinated quasi modules As an example, family quantum deforming the associated to non-degenerate even l...

The growth function of a graph with respect to a vertex is near polynomial if there exists a polynomial bounding it above for infinitely many positive integers. In the paper vertex-symmetric undirected graphs and vertex-symmetric directed graphs with coinciding inand out-degrees are described in the case their growth functions are near polynomial.

We consider the full multinomial combinatorics of all irreducible representations of the octahedral (cubic) symmetry as a function of partitions for vertex, face and edge colorings. Full combinatorial tables for all irreducible representations and all multinomial partitions are constructed. These enumerations constitute multinomial expansions of character-based cycle index polynomials, and grow...