Some Metrical Properties of Lattice Graphs of Finite Groups

: This paper is concerned with the combinatorial facts of the lattice graphs of Z p 1 × p 2 ×···× p m , Z p m 1 1 × p m 2 2 , and Z p m 1 1 × p m 2 2 × p 13 . We show that the lattice graph of Z p 1 × p 2 ×···× p m is realizable as a convex polytope. We also show that the diameter of the lattice graph of Z p m 1 1 × p m 2 2 ×···× p mrr is r ∑ i = 1 m i and its girth is 4.

, and Z p m Keywords: finite group; lattice graph; convex polytope; diameter; girth

Introduction
The relation between the structure of a group and the structure of its subgroups constitutes an important domain of research in both group theory and graph theory. The topic has enjoyed a rapid development starting with the first half of the twentieth century.
The main object of this paper is to study the interplay of group-theoretic properties of a group G with graph-theoretic properties of its lattice graph L(G). Every group has a corresponding lattice graph, which can be finite or infinite depending upon the order of the group. This study helps illuminate the structure of the set H(G) of subgroups of G.
The lattice graph L(G) of a finite cyclic group G is obtained as follows: Each vertex of L(G) corresponds to an element of H(G), and two vertices corresponding to two elements H 1 , H 2 of H(G) are connected by an edge if and only if H 1 ≤ H 2 and that there is no element K of H(G) such that H 1 K H 2 (see [1,2]), thus ≤ is used when H 1 is proper maximal subgroup of H 2 . The notation ≤ is used as subgroup. Degree of a vertex is the number of edges attached to that vertex. A vertex is defined to be even or odd if its degree is even or odd. The degree vector of a graph is the sequence of degrees of its vertices arranged in non-increasing order [3]. The diameter diam(G) of a connected graph G is the maximum distance among vertices of G [4,5]. The girth g(G) of a graph G is the length of a smallest cycle in G, and is infinity if G is acyclic [6]. It is a fact that the lattice of subgroups of a given group can rarely be drawn without its edges crossing [1,2]. The crossing number cr(G) of a graph G is the minimum number of crossings of its edges among the drawings of G in the plane. A graph is considered Eulerian if there exists a Eulerian path in which we can start at a vertex, traverse through every edge only once, and return to the same vertex where we started. A connected graph G is Eulerian if each vertex has even degree, and is semi-Eulerian if it has exactly two vertices with odd degrees [7].
A polytope is a finite region of R n enclosed by a finite number of hyper-planes. A polytope is called convex if its points form a convex subset of R n . Combinatorial aspects of the groups can be computed using its lattice graphs [8,9]. The authors discussed some finite simple groups of low rank in [8]. Tarnauceanu introduced new arithmetic method of counting the subgroups of a finite Abelian group in [10]. The author of [11] described the finite groups Ghaving |G| − 1 cyclic subgroups. Saeedi and Farrokhi [12] computed factorization number of some finite groups. Tarnauceanu [13] characterized elementary Abelian two-groups. Tarnauceanu and Toth discussed cyclicity degree of finite groups in [14]. Tarnauceanu [15] discussed finite groups with dismantlable subgroup lattices.
In the present article, we are interested in the lattices of finite groups. We demonstrate that the lattice graph of Z p 1 ×p 2 ×···×p m can be viewed as a convex polytope, and that the diameter of the lattice graph of Z p m 1 1 ×p m 2 2 ×···×p mr r is r ∑ i=1 m i and its girth is 4. We also compute many other properties of these lattices. We are interested developing some combinatorial invariants of the groups coming from their lattice graphs. The main motivation comes from the fact that finite graphs are relatively easier to handle than the finite groups. One is interested in capturing facts of the group while studying the graph associated to it. This study ultimately relates some parameters of the graphs with some parameters of the groups. Several authors computed some combinatorial aspects of finite graphs such as diameter, girth and radius but, for the lattice graph, similar questions are still open and need to be addressed. This article can be considered as a step forward in this direction. We, naturally, pose problems about the exact values of these parameters for more general classes of groups such as S n , A n , and sporadic groups.

The Results
In this section, we give the combinatorial results about the lattice graphs of Z p 1 ×p 2 ×···×p m , In the end, we give the general results about L(Zn).

82
Now the maximal chains: The subgroup H 0 =< 0 > is contained immediately 83 in m subgroups of Z n : H 11 =< p 1 × p 2 × · · · × p m−1 >, H 12 =< p 1 × p 2 × · · · × 84 p m−2 × p m >, H 1m =< p 2 × p 3 × · · · × p m >, as you can see in the        Each H 1 i is contained in m − 1 subgroups of Z n : For instance, Figure 2).     Similarly, every H 2 j is contained in m − 2 subgroups of Z n : Figure 3).   The process will continue till we receive < p 1 >, < p 2 > . . . < p m > at the 96 second last stage. Now each one of these is contained in < 1 > = Z n and the  The process will continue until we receive < p 1 >, < p 2 > . . . < p m > at the second last stage. Now, each one of these is contained in < 1 > = Z n and the process is finished (see Figure 4).   The process will continue till we receive < p 1 >, < p 2 > . . . < p m > at the we put all these chains of subgroup in the plane such that each subgroup is identified to itself occurring in all these series. Thus, the same subgroups that appear in more than one series, appear only as a single vertex of the lattice graph in the plane. This lattice graph starts of at the identity and finishes at Z n because these subgroups appear in all series. This may have crossings. If we imagine this gluing in R 3 avoiding all crossings in higher dimension, we get a convex polytope with 2 m vertices, one vertex corresponding to each element of H(Z n ). For instance, the cases for m = 3, 4, 5 are shown in Figures 5-7

Proof. (a)
It is clear that Z n is cyclic, thus, for each divisor, we have a unique subgroup. Prime divisor p i of n yields maximum quotient, thus these numbers correspond to the maximal subgroups of Z n , which are < p 1 >, < p 2 >, < p 3 >, . . . , < p m−1 >, < p m >. to < 1 >; see, for instance, a typical series: (c) diamL(Z n ) is one less than the length of a maximal series, which is the length of each path from < 0 > to < 1 >. (d) At the first stage, the degree of the vertex H 0 is m because it is adjacent to m vertices H 1 i , i = 1, 2, . . . , m in the second stage, as shown in the construction of L(Z n ). The degree of each vertex at the second stage is m as each vertex H 1 i is adjacent to m − 1 vertices lying higher to it and to one vertex lying below it. For instance, Continuing this process, we receive that the vertices corresponding to < p 1 >, < p 2 >, . . . , < p m > are adjacent to the vertex corresponding to < 1 >, giving the degree of the last vertex m too. Thus, each vertex of L(Z n ) has degree m, and the proof is finished.

Remark 1.
It should be remarked that the lattice graph of Z n is a small-world network in which nearly all nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other and most nodes can be reached from every other node by a small number of steps. Thus, it is a connected graph [16].  . The process will continue until we receive a subgroup < p 1 × p m 2 2 > that is contained in next two subgroups, < p 1 × p m 2 2 > and < p m 2 2 >. Both are further contained in < p m 2 −1 2 >, which itself is contained in < p m 2 −2 2 >. This process is continued until we obtain a subgroup < p 2 >, which is continued in < 1 > = Z ).
(b) Again, this is obvious; see a typical maximal series: In this case, the diameter is exactly one less than the number of elements in a maximal series. as shown in Figure 8.
. The process will continue till we receive a subgroup >. Continuing this process till we obtain < p 2 > which is contained in < 1 > = Z p m 1 1 ×p ; for instance, < 0 >⊆< p m 1 −1 Similarly, all the remaining subgroups will form other series of subgroups. At the end, gluing the vertices occurring in all maximal series, we obtain the required graph, as shown in Figure 9.  Proof. The proof is clear from the Table 1: >⊆ . . . ⊆< p 2 >⊆< 1 >, which consists of m 1 + m 2 + 2. Since each maximal series has the same number of elements, we are done. Proof. The proof is given in the following Table 2:

Conclusions
In this article, we discuss some metrical aspects of the lattice graphs of some families of finite groups. We obtain the diameter and girth as well as many other aspects of these lattice graphs. Along with many other results, the following are the main contributions of this article.