Detour spectrum and detour energy of conjugate graph complement of dihedral group

C C Marzuki,Muzakir Muzakir,Abdussakir Abdussakir

doi:10.1088/1742-6596/1028/1/012111

C C Marzuki, Muzakir Muzakir + Show 1 more

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https://doi.org/10.1088/1742-6596/1028/1/012111

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Abstract

Study of graph from a group has become an interesting topic until now. One of the topics is spectra of a graph from finite group. Spectrum of a finite graph is defined as collection of all distinct eigenvalues and their algebraic multiplicity of its matrix. The most related topic in the study of spectrum of finite graph is energy. Energy of a finite graph is defined as sum of absolute value of all its eigenvalues. In this paper, we study the spectrum and energy of detour matrix of conjugate graph complement of dihedral group. The main result is presented as theorems with complete proof.

Highlights

IntroductionSeveral graphs from some group have been studied by researchers, such as Cayley graph [1,2], Schreier coset graph [3], identity graph [4], commuting [5,6] and non-commuting graph[7,8,9], subgroup graph [10,11], power graph [12], inverse graph [13,14] and conjugate graph [15] of a group
Spectrum of a finite graph is defined as collection of all distinct eigenvalues and their algebraic multiplicity of its matrix
Energy of a finite graph is defined as sum of absolute value of all its eigenvalues

Summary

Introduction

Several graphs from some group have been studied by researchers, such as Cayley graph [1,2], Schreier coset graph [3], identity graph [4], commuting [5,6] and non-commuting graph[7,8,9], subgroup graph [10,11], power graph [12], inverse graph [13,14] and conjugate graph [15] of a group. The conjugate graph of group G contains all elements of G as its vertex set and two distinct vertices will be adjacent if they are representatives of the same conjugacy class [15]. Conjugate graph of a group G will be denoted by C(G) and the complement of C(G) will be denoted by C(G). Detour matrix of graph G of order p that denoted by DD(G) is a (p × p)-matrix DD(G) = (Dij ). Let λi1 > λi2 > λi3 > ⋯ > λin are the distinct eigenvalues of DD(G), the spectrum of DD(G) can be written as specDD (G) =. Since the study of detour spectrum and energy of conjugate graph complement of dihedral group has not been done yet, we do this study

Result

Conclusion