Some Metrical Properties of Lattice Graphs of Finite Groups

Jia-Bao Liu,Abdul Rauf Nizami,Qurat-Ul-Ain Munir,Mobeen Munir

doi:10.3390/math7050398

Abstract

This paper is concerned with the combinatorial facts of the lattice graphs of Z p 1 × p 2 × ⋯ × p m , Z p 1 m 1 × p 2 m 2 , and Z p 1 m 1 × p 2 m 2 × p 3 1 . 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 1 m 1 × p 2 m 2 × ⋯ × p r m r is ∑ i = 1 r m i and its girth is 4.

Highlights

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
We show that the lattice graph of Z p1 × p2 ×···× pm is realizable as a convex r polytope
We show that the diameter of the lattice graph of Z pm1 × pm2 ×···× pmr is ∑ mi and its r i =1 girth is 4

Summary

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 lattice graph L( G ) of a finite cyclic group G is obtained as follows: Each vertex of L( G ). It is a fact that the lattice of subgroups of a given group can rarely be drawn without its edges crossing [1,2]. Tarnauceanu and Toth discussed cyclicity degree of finite groups in [14]. Tarnauceanu [15] discussed finite groups with dismantlable subgroup lattices. We are interested developing some combinatorial invariants of the groups coming from their lattice graphs. 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. Pose problems about the exact values of these parameters for more general classes of groups such as Sn , An , and sporadic groups

The Results

Conclusions