T. Tamizh Chelvam and T. Asir [15] studied the intersection graph of gamma sets in the total graph of a commutative Artin ring. The intersection graph ITΓ(R) of gamma sets in the total graph of a commutative ring R, is the undirected graph with vertex as collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v 6= ∅. In this paper, we c...

Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents.

Motivated by the Eulerian ribbon graph minors, in this paper we introduce notion of checkerboard colourable minors for graphs and its dual: bipartite graphs. abstract graphs, another i.e. join are also introduced. Using these then give excluded minor characterizations classes plane

In this paper, a hybrid Lagrangian–Eulerian topology optimization (LETO) method is proposed to solve the elastic force equilibrium with Material Point Method (MPM). LETO transfers density information from freely movable Lagrangian carrier particles fixed set of Eulerian quadrature points. This transfer based on smooth radial kernel involved in compliance objective avoid artificial checkerboard ...

‎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).‎    

AbstractThe notion of the prime graph a ring R was first introduced by Bhavanari, Kuncham, and Dasari in 2010. The R, denoted PG(R) is whose vertices are all elements ring, where two distinct x y adjacent if only xRy = 0 or yRx 0. In this paper, we study forms properties integer modulo group, some examples number its spanning trees. it found that for n, maximum degree PG(Z_n) n-1 minimum 1. The...

The odd edge connectivity of a graph G, denoted by o(G), is the size of a smallest odd edge cut of the graph. Let S be any given surface and be a positive real number. We proved that there is a function fS( ) (depends on the surface S and lim !0 fS( )1⁄41) such that any graph G embedded in S with the odd-edge connectivity at least fS( ) admits a nowhere-zero circular (2þ )-flow. Another major r...

We prove a new sufficient condition for a cubic 3-connected planar graph to be Hamiltonian. This condition is most easily described as a property of the dual graph. Let G be a planar triangulation. Then the dual G∗ is a cubic 3-connected planar graph, and G∗ is bipartite if and only if G is Eulerian. We prove that if the vertices of G are (improperly) coloured blue and red, such that the blue v...

The study of genome rearrangement has many flavours, but they all are somehow tied to edit distances on variations of a multi-graph called the breakpoint graph. We study a weighted 2-break distance on Eulerian 2-edge-colored multi-graphs, which generalizes weighted versions of several Double Cut and Join problems, including those on genomes with unequal gene content. We affirm the connection be...

The harmonious coloring of an undirected simple graph is a vertex coloring such that adjacent vertices are assigned different colors and each pair of colors appears together on at most one edge. The harmonious chromatic number of a graph is the least number of colors used in such a coloring. The harmonious chromatic number of a path is known, whereas the problem to find the harmonious chromatic...