In this paper, we show that for the alternating group An, the class C of n- cycle, CC covers An for n when n = 4k + 1 > 5 and odd. This class splits into two classes of An denoted by C and C/, CC= C/C/ was found.

     Let G be a finite group and X be a G-conjugacy of elements of order 3. The A4-graph of G is a simple graph with vertex set X and two vertices x,yÎX are linked if x≠ y and xy^-1 is an involution element. This paper aims to investigate  the A4-graph properties for the monster  Held group He.

Assume that G is a finite group and X = tG where t is non-identity element with t3 = 1. The simple graph with node set being X such that a, b ∈ X, are adjacent if ab-1 is an involution element, is called the A4-graph, and designated by A4(G, X). In this article, the construction of A4(G, X) is analyzed for G is the twisted group of Lie type 3D4(3).

Foundations supporting reciprocating engines, radar towers, turbines, large electric motors, and generators, etc. are subject to vibrations caused by unbalanced machine forces as well as the static weight of the machine. If these vibrations are excessive, they may damage the machine or cause it not to function properly. In the case of block foundation, if changes in size and mass of the foundation do not lead to a satisfactory design, a pile foundation may be used. In this study, the dynamic response of piles and pile Groups in dry sand is investigated experimentally. The analysis involves the displacement response under harmonic excitation. In addition, a numerical modeling by using finite element method with a three-dimensional formula

     This work aims to introduce and to study a new kind of divisor graph which is  called idempotent divisor graph, and it is  denoted by . Two non-zero distinct vertices v₁ and v₂ are adjacent if and only if , for some non-unit idempotent element . We establish some fundamental properties of ,  as well as it’s connection with . We also study planarity of this graph.

In the present paper, we have introduced some new definitions On D- compact topological group and D-L. compact topological group for the compactification in topological spaces and groups, we obtain some results related to D- compact topological group and D-L. compact topological group.

In the present paper, we have introduced some new definitions On D- compact topological group and D-L. compact topological group for the compactification in topological spaces and groups, we obtain some results related to D- compact topological group and D-L. compact topological group.

     Let  be any connected graph with vertices set  and edges set .  For any two distinct vertices  and , the detour distance between  and  which is denoted by  is a longest path between  and  in a graph . The detour polynomial of a connected graph  is denoted by ;  and is defined by . In this paper, the detour polynomial of the theta graph and the uniform theta graph will be computed.

A graph
is said to be singular if and only if its adjacency matrix is singular. A graph
is said to be bipartite graph if and only if we can write its vertex set as
, and each edge has exactly one end point in
and other end point in
. In this work, we will use graphic permutation to find the determinant of adjacency matrix of bipartite graph. After that, we will determine the conditions that the bipartite graph is singular or non-singular.