The Energy of Cayley Graphs for Symmetric Groups of Order 24

ASM Sc. J., 13, 2020 https://doi.org/10.32802/asmscj.2020.sm26(1.22)

2 3 Amira Fadina Ahmad Fadzil1*, Nor Haniza Sarmin and Ahmad Erfanian

1,2 Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia. 3Department of Pure Mathematics, Faculty of Mathematical Science, and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran.

A Cayley graph of a finite group G with respect to a subset S of G is a graph where the vertices of the graph are the elements of the group and two distinct vertices x and y are adjacent to each other if xy−1 is in the subset S. The subset of the Cayley graph is inverse closed and does not include the identity of the group. For a simple finite graph, the energy of a graph can be determined by summing up the positive values of the eigenvalues of the adjacency matrix of the graph. In this paper, the graph being studied is the Cayley

graph of symmetric group of order 24 where S is the subset of S4 of valency up to two. From the Cayley graphs, the eigenvalues are calculated by constructing the adjacency matrix of the graphs and by using some properties of special graphs. Finally, the energy of the respected Cayley graphs is computed and presented. Keywords: energy of graph; cayley graph; symmetric groups

I. INTRODUCTION Konstantinova in 2008, has surveyed the historical changes of some problems on Cayley graphs such as the Hamiltonicity The study on Cayley graphs was initiated by Arthur Cayley in and diameter problems. The author also included various 1878 to explain the idea of abstract groups described by a set uses of Cayley graphs in solving combinatorial, graph of generators (Ghorbani & Nowroozi-Larki, 2018). The theory theoretical and applied problems. Furthermore in 2012, has been advanced into a significant branch in algebraic Adiga and Ariamanesh have determined the number of graph theory. There are many problems regarding Cayley undirected Cayley graphs of symmetric group and alternating graphs that have been studied by many graph theorists and groups up to isomorphism. group theorists. In addition, the study on the energy of general simple In 1988, Babai and Seress have studied on the diameter of graphs was first defined by Gutman in 1978 inspired from the the Cayley graphs of symmetric groups and the alternating Huckel Molecular Orbital (HMO) theory proposed in 1930s. groups. Not long after, in 1993, Lakshmivarahan et. al., have The theory has been used by chemists in approximating the analysed the symmetries in the interconnection networks of energies related with π-electron orbitals in conjugated a variety of Cayley graphs of permutation groups. The types hydrocarbon. Besides, its chemical applications, there are a of symmetry analysed consist of vertex and edge transitivity, few applications in other field of science such as in graph distance regularity and distance transitivity entropies (Dehmer et. al., 2015), modelling of properties of Meanwhile in 2000, Friedman has shown in his study that proteins (Wu et. al., 2015) and also was used in the search for among all sets of n - 1 transpositions which generate the the genetic causes of Alzheimer disease (Daianu et al., 2015). symmetric groups, the Cayley graph associated to set 푆 = Therefore, this paper aimed to present the energy of the {(1, 푛), (2, 푛), . . . , (푛 − 1, 푛)} as the highest eigenvalues. Cayley graphs associated to symmetric group of order 24 with

______*Corresponding author’s e-mail: [email protected]

ASM Science Journal, Volume 13, 2020 respect to the subsets S of valency one and two. The Definition 3 (Beineke & Wilson, 2004) Complete Graph methodology consists of listing the subsets S, constructing A complete graph 퐾푛 has 푛 vertices, each of which is adjacent the Cayley graphs with respect to the subsets S, finding their to all of the others. isomorphism, building the adjacency matrix for the graphs, calculating the eigenvalues of the adjacency matrix of the Definition 4 (Pemmaraju & Skiena, 2003) Cycle Graph graphs and computing the energy of the Cayley graphs. A cycle graph 퐶푛 , sometimes simply known as an 푛-cycle, is a This paper is structured as follows: in Section 1, the early graph on 푛 vertices containing a single cycle through all the studies for the topics are introduced while in Section 2, the vertices. preliminaries results that are being used for this study are presented. In Section 3, the main results are discussed in two Proposition 1 (Brouwer & Haemers, 2011) Spectrum of propositions and theorems and finally, Section 4 gives the Complete Graph conclusion to the main results. Let be the complete graph 퐾푛 on 푛 vertices. Its spectrum is {(푛 − 1)1, ( − 1)푛 − 1 }. II. PRELIMINARIES Proposition 2 (Brouwer & Haemers, 2011) Spectrum of The followings are some basic concepts and properties that Cycle Graph are used in this study. Consider the undirected 푛-cycle graph 퐶푛 . The spectrum of

퐶푛 consists of the numbers {2 푐표푠(2휋푗⁄푛); 푗 = {0,1, … , 푛 − 1}. Definition 1 (Robinson, 1996) Symmetric Group If 푋 is a nonempty set, a bijection 휋: 푋 → 푋 is called a Definition 5 (Bapat, 2010) Adjacency Matrix permutation of 푋. The set of all permutations of 푋 is a group Let 훤 be a graph with 푉(훤) = {1, … , 푛} and 퐸(훤) = {푒1, … , 푒푚}. with respect to functional composition called the symmetric The adjacency matrix of 훤 denoted by 퐴(훤) is the 푛 × 푛 matrix groups on 푋, 푆푦푚 푋. When 푋 = { 1 ,2 , … , 푛}, it is customary to defined as follows. The rows and the columns of 퐴(훤) are write 푆푛 for 푆푦푚 푋 and to call this the symmetric groups of indexed by 푉(훤). If 푖 ≠ 푗 then the (푖, 푗) −entry of 퐴(훤) is 0 for degree 푛. This study focused on the symmetric group of order vertices 푖 and 푗 nonadjacent, and the (푖, 푗) −entry is 1 for 푖 and

24 denoted as 푆4. The elements of 푆4 = 푗 adjacent. The (푖, 푖) −entry of 퐴(훤) is 0 for 푖 = 1, … , 푛. 퐴(훤) {(1), (12), (13), (14), (23), (24), (34), (123), is often simply denoted by 퐴. (132), (124), (142), (134), (143), (234), (243), (1234), (1432), (1243), (1342), (1324), (1423), Definition 6 (Bapat, 2010) Energy of Graph (12)(34), (13)(24), (14)(23)}. This study also focused on For any graph 훤, the energy of the graph is defined as 휀(Γ) = 푛 the subsets of 푆4 of valency 1 and 2 where the valency of the ∑푖=1 |휆푖| where 휆1, … , 휆푛 are the eigenvalues of the adjacency subsets is the degree of the subsets. matrix of 훤.

Definition 2 (Beineke & Wilson, 2004) Cayley Graph of a III. MAIN RESULTS Group Let 퐺 be a finite group with identity 1. Let 푆 be a subset of 퐺 In this section, our main results are specified in terms of satisfying 1 ∉ 푆 and 푆 = 푆−1; that is, 푠 ∈ 푆 if and only if 푠−1 ∈ propositions and theorems. The propositions present the

푆. The Cayley graph 퐶푎푦(퐺; 푆) on 퐺 with connection set 푆 is generalization of the Cayley graphs of 푆4 with respect to the defined as follows: subsets 푆 of valency one and two. Then, the results on the (i) the vertices are the elements of 퐺 energy of the Cayley graphs found are presented in two (ii) there is an edge joining 푔 and ℎ if and only if ℎ = 푠푔 theorems. The findings in the propositions are used in for some 푠 ∈ 푆. proving the theorems following them. The set of all Cayley graphs on 퐺 is denoted by 퐶푎푦(퐺; 푆) where 푆 is the subset of 퐺 with a certain valency which is the Proposition 3 (1) order of 푆. Let 푆4 be the symmetric group of order 24 and 푆 be the (1) subset of 푆4 of valency one, |푆 | = 1. Then, the Cayley graph

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(1) (1) of 푆4 with respect to the subset 푆 , denoted as 퐶푎푦(푆4, 푆 ) Consider the symmetric group of order 24, 푆4. By Proposition 12 (1) 12 is ⋃푖=1 퐾2. 3, the Cayley graph, 퐶푎푦(푆4, 푆 ) = ⋃푖=1 퐾2. By Proposition 1,

the adjacency spectrum of a complete graph, 퐾푛 is Proof {(푛 – 1)1, ( − 1)푛 – 1 }. Thus, the adjacency spectrum of Suppose 푆 is the symmetric group of order 24 and 12 1 2−1 4 ⋃푖=1 퐾2 is 12{(2 – 1) , (−1) }. This gives the eigenvalues, (1) 퐶푎푦(푆4, 푆 ) is the Cayley graph of 푆4 with respect to the (1) 휆푖 = ±1 with multiplicity 12. Therefore, 퐸 (퐶푎푦(푆4, 푆 )) = subset 푆(1) of valency one. The vertex set of the Cayley graph, 12(1) + 12(|−1|) = 24. (1) 푉 (퐶푎푦(푆4, 푆 )) = {(1), (12), (13), (14), (23), (24), (34), (123), (132), (124), (142), (134), (143), Theorem 2 (2) (234), (243), (1234), (1432), (1243), (1342), Let 푆4 be the symmetric group of order 24 and 푆 be the

(1324), (1423), (12)(34), (13)(24), (14)(23)}. subset of 푆4 of valency two. Then, the energy of the Cayley (1) (2) Then, by the definition of Cayley graph, let 푆 = {(12)(34)}, graph of 푆4 with respect to the subset 푆 , denoted as (1) (2) the edge set of the Cayley graph, 퐸푑푔푒(퐶푎푦(푆_4, 푆 )) = 퐸(퐶푎푦(푆4, 푆 )) is given in the following: { {(1), (12)(34)}, 32 ; 푆 = {(푖푗), (푘푙)} 푎푛푑 {(푖푗푘), (푙푚푛)},

; 푆 = {(푖푗), (푘푙)(푚푛)}, {(12), (34)}, {(13), (1432)}, {(14), (1342)}, 12 + 12√2 24 ; 푆 = {(푖푗), (푖푗)(푘푙)}, {(푖푗)(푘푙), (푚푛)(푝푞)} {(23), (1243)}, {(24), (1234)}, {(123), (243)}, { 푎푛푑 {(푖푗푘푙), (푚푛푝푞)}. {(132), (143)}, {(124), (234)}, {(142), (134)}, {(1324), (1423)}, {(13)(24), (14)(23) }}. Proof (1) Thus, 퐶푎푦(푆4, 푆 ) is the union of 12 complete graphs of Consider the symmetric group of order 24, 푆4. By Proposition 12 (2) two vertices, ⋃푖=1 퐾2. ∎ 4, 퐶푎푦(푆4, 푆 ) is given in the following: 4 ⋃ 퐶 ; 푆 = {(푖푗), (푘푙)}, Proposition 4 6 푖=1 (2) 3 Let 푆4 be the symmetric group of order 24 and 푆 be the

(2) ⋃ 퐶8 ; 푆 = {(푖푗), (푘푙)(푚푛)}, subset of 푆4 of valency two, |푆 | = 2. Then, the Cayley graph (2) 푖=1 (2) (2) 퐶푎푦(푆4, 푆 ) = 8 of 푆4 with respect to the subset 푆 , denoted as 퐶푎푦(푆4, 푆 ) ⋃ 퐶3 ; 푆 = {(푖푗푘), (푙푚푛)}, is given in the following: 푖=1 4 6 ; 푆 = {(푖푗), (푖푗)(푘푙)},

⋃ 퐶 ; 푆 = {(푖푗), (푘푙)}, ⋃ 퐶 {(푖푗)(푘푙), (푚푛)(푝푞)} 6 4 푖=1 { 푖=1 푎푛푑 {(푖푗푘푙), (푚푛푝푞)}. 3

⋃ 퐶 ; 푆 = {(푖푗), (푘푙)(푚푛)}, 8 (2) 푖=1 From Proposition 2, the adjacency spectrum of a cycle 퐶푎푦(푆4, 푆 ) = 8 graph 퐶푛 is {2 푐표푠(2휋푗⁄푛); 푗 = {0,1, … , 푛 − 1}. Thus, the ⋃ 퐶3 ; 푆 = {(푖푗푘), (푙푚푛)}, (2) 푖=1 adjacency spectrum of 퐶푎푦(푆4, 푆 ) is given in the following: 6 ; 푆 = {(푖푗), (푖푗)(푘푙)}, 4 ⋃ 퐶 {(푖푗)(푘푙), (푚푛)(푝푞)} 2휋푗 4 4 {2 푐표푠( )} ; ⋃ 퐶6 , 푎푛푑 {(푖푗푘푙), (푚푛푝푞)}. 6 { 푖=1 푖=1 3 2휋푗 3 {2 푐표푠( )} ; ⋃ 퐶 , Proof 8 8 (2) 푖=1 Spec(퐶푎푦(푆4, 푆 )) = 8 The proof is similar to the proof in the previous proposition. 2휋푗 8 {2 푐표푠( )} ; ⋃ 퐶3 , 3 푖=1 6 Theorem 1 2휋푗 (1) 6 {2 푐표푠( )} ; ⋃ 퐶4 . Let 푆4 be the symmetric group of order 24 and 푆 be the 4 { 푖=1 subset of 푆 of valency one. Then, the energy of the Cayley 4 (1) graph of 푆4 with respect to the subset 푆 , denoted as (2) Therefore, the energy of the Cayley graphs, 퐸(퐶푎푦(푆4, 푆 )) (1) 퐸(퐶푎푦(푆4, 푆 ) ) is 24. is given in the following: Proof

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32 ; 푆 = {(푖푗), (푘푙)} 푎푛푑 {(푖푗푘), (푙푚푛)},

; 푆 = {(푖푗), (푘푙)(푚푛)}, 12 + 12√2 24 ; 푆 = {(푖푗), (푖푗)(푘푙)}, {(푖푗)(푘푙), (푚푛)(푝푞)} { 푎푛푑 {(푖푗푘푙), (푚푛푝푞)}. ∎

Example (1) Let 푆4 be the symmetric group of order 24 and 푆 = {(13)} be the subset of 푆4. Then, the Cayley graph of 푆4 with respect (1) 12 to the subset 푆 , denoted as 퐶푎푦(푆4, {(13)}) is ⋃푖=1 퐾2.

Therefore, the energy of the Cayley graph, 퐸(퐶푎푦(푆4, {(13)})) is 24.

Proof (1) Consider the symmetric group of order 24, 푆4 and 푆 =

{(13)} be the subset of 푆4. Then, by Definition 2, the vertex 푥 is connected to 푠푥 where 푠 ∈ 푆.

(1) − (13) since (13)(1) = (13) From the adjacency matrix, the characteristic polynomial is (12) − (123) since (13)(12) = (123) found to be 푓(휆) = 휆24 − 12휆22 + 66휆20 − 220휆18 + 495휆16 − (14) − (143) since (13)(14) = (143) 792휆14 + 924휆12 − 792휆10 + 495휆8 − 220휆6 + 66휆4 − 12휆2 + (23) − (132) since (13)(23) = (132) 1. (24) − (13)(24) since (13)(24) = (13)(24) By using the generalization of spectrum of a complete (34) − (134) since (13)(34) = (134) graph, the eigenvalues can be found as ±1 with multiplicity (124) − (1243) since (13)(124) = (1243) 12. Therefore, the energy of the Cayley graph, (142) − (1423) since (13)(142) = (1423) 퐸(퐶푎푦(푆4, {(13)})) = 12(1) + 12(|−1|) = 24. ∎ (1234) − (12)(34) since (13)(1234) = (12)(34)

(1324) − (243) since (13)(1324) = (243) (1342) − (234)since (13)(1342) = (234) IV. SUMMARY (1432) − (14)(23) since (13)(1432) = (14)(23). As a conclusion, the energy of the Cayley graphs for the The connected elements form the 퐶푎푦(푆4, {(13)}) is 12 symmetric group of order 24 with respect to the subsets 푆 of ⋃푖=1 퐾2as illustrated in the following figure. valency one and two have been found and are summarized as in the following table.

Table 1. Energy of Cayley graphs of 푆4

|푆| Subset 푆 퐶푎푦(푆4, 푆) Energy 12

1 {(푖푗)}, {(푖푗)(푘푙)} ⋃ 퐾2 24 푖=1 4

{(푖푗), (푘푙)} ⋃ 퐶6 32 푖=1 6

{(푖푗), (푖푗)(푘푙)} ⋃ 퐶4 24 Figure 1. 퐶푎푦(푆4, {(13)}) 푖=1 2 3 12 {(푖푗), (푘푙)(푚푛)} ⋃ 퐶 By the definition of the adjacency matrix, 8 푖=1 + 12√2 퐴(퐶푎푦(푆4, {(13)})) = 6

{(푖푗)(푘푙), (푚푛)(푝푞)} ⋃ 퐶4 24 푖=1

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{(푖푗푘), (푙푚푛)} ⋃ 퐶3 32 푖=1 6

{(푖푗푘푙), (푚푛푝푞)} ⋃ 퐶4 24 푖=1

V. ACKNOWLEDEGMENTS

The authors would like to express their appreciation to Malaysian Mathematical Science Society (PERSAMA) for the support of sponsor and Research Management Centre (RMC), Universiti Teknologi Malaysia (UTM) Johor Bahru for the financial funding through the Research University Grant (GUP) Vote No. 13J82. The first author would also like to express her appreciation to UTM for the financial support under Zamalah Scholarship.

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VI. REFERENCES

Adiga, C., & Ariamanesh, H. 2012, Some properties of Cayley Ghorbani, M., & Nowroozi-Larki, F. 2018, On the Spectrum

graphs on symmetric groups Sn. International Journal of of Cayley Graphs Related to the Finite Groups. Filomat, Algebra, 6(17), 807-813. 31(20). Babai, L., & Seress, A. 1988, On the diameter of Cayley graphs Gutman, I. 1978, The Energy of Graph. Der. Math. Stat. Sekt. of the symmetric group. Journal of combinatorial theory, Forschungszent Graz., 103, 1-22. Series A, 49(1), 175-179. Konstantinova, E. 2008, Some problems on Cayley graphs. Bapat, R. B. 2010, Graphs and matrices. New York (NY): Linear Algebra and its Applications, 429(11), 2754-2769. Springer. Lakshmivarahan, S., Jwo, J. S., & Dhall, S. K. 1993, Symmetry Beineke, L.W., & Wilson, R. J. 2004, Topics in algebraic in interconnection networks based on Cayley graphs of graph theory. Vol. 102. USA: Cambridge University Press. permutation groups: A survey. Parallel Computing, 19(4), Brouwer, A. E., & Haemers, W. H. 2011, Spectra of graphs. 361407. Springer Science and Business Media. Pemmaraju, S., & Skiena, S. 2003, Computational Discrete Daianu, M., Mezher, A., Jahanshad, N., Hibar, D. P., Nir, T. Mathematics: Combinatorics and Graph Theory with M., Jack, C. R., Weiner, M. W., Bernstein, M. A., & Mathematica. Cambridge University Press. Thompson, P. M. 2015, Spectral graph theory and graph Robinson, D. J. 1996, A course in the theory of groups, 2nd. energy metrics show evidence for the Alzheimers disease ed. USA: Springer-Verlag New York, Inc. disconnection syndrome in APOE-4 risk gene carriers. Wu, H., Zhang, Y., Chen, W., & Mu, Z. 2015, Comparative Proc. IEEE Int. Symp. Biomed Imaging 2015, 458461. analysis of protein primary sequences with graph energy. Dehmer, M. Li, X., & Shi, Y. 2015, Connections between Physica A: Statistical Mechanics and its Applications, 437, generalized graph entropies and graph energy. Complexity, 249262. 21, 3541. Friedman, J. 2000, On Cayley graphs on the symmetric group generated by tranpositions. Combinatorica, 20(4), 505-519.