the order graph of a group $g$, denoted by $gamma^*(g)$, is a graph whose vertices are subgroups of $g$ and two distinct vertices $h$ and $k$ are adjacent if and only if $|h|big{|}|k|$ or $|k|big{|}|h|$. in this paper, we study the connectivity and diameter of this  graph. also we give a relation between the order graph and prime  graph of a group.

let f, g and h be non-empty graphs. the notation f → (g,h) means that if any edge of f is colored by red or blue, then either the red subgraph of f con- tains a graph g or the blue subgraph of f contains a graph h. a graph f (without isolated vertices) is called a ramsey (g,h)−minimal if f → (g,h) and for every e ∈ e(f), (f − e) 9 (g,h). the set of all ramsey (g,h)−minimal graphs is denoted by ...

in this paper, we first collect the earlier results about some graph operations and then wepresent applications of these results in working with chemical graphs.

We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial graphs on $n$~vertices. A component maximal set of vertices supports both spanning tree. show there are critical pairs edge densities at which giant appears. In contrast to standard graph model, phase transition first order: size largest jumps from $O(1)$ $\Theta(n)$ point. connect thi...

Proper coloring of the vertices of a graph with minimum number of colors has always been of great interest of researchers in the field of soft computing. Genetic Algorithm (GA) and its application as the solution method to the Graph Coloring problem have been appreciated and worked upon by the scientists almost for the last two decades. Various genetic operators such as crossover and mutation h...

The initial phase of the inspiral process a binary system producing gravitational waves can be described by perturbation theory. At third post-Minkowskian order two-loop double box graph, known as H-graph contributes. We consider case where two objects making up have equal masses. express all master integrals related to equal-mass weight four in terms multiple polylogarithms. provide numerical ...

This is a review of my recent work. What kinds of combinations of singularities can appear in small deformation fibers of a fixed singularity? We consider this problem for hypersurface singularities on complex analytic spaces of dimension 2. Recall first that a connected Dynkin graph of type A, D or E corresponds to a surface singularity called a rational double point. (Durfee [3].) When the fi...

A cycle double cover (CDC) of an undirected graph is a collection of the graph’s cycles such that every edge of the graph belongs to exactly two cycles. We describe a constructive method for generating all the cubic graphs that have a 6-CDC (a CDC in which every cycle has length 6). We also prove that all such graphs have a Hamiltonian cycle.

We study the number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths, in which the tiles are “vertically decreasing”. We use these quantities to compute pairing probabilities in the double-dimer model: Given a planar bipartite graph G with special vertices, called nodes, on the outer face, the doubledimer model is formed by the superposition of a uniformly random dimer c...

To every directed graph G one can associate a complex (G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn, where it leads to studying some interesting representations of the symmetric group and corresponds (via Stanley-Reisner correspondence) to an interesting quotient ring. Our main ...