'- :"'\r'ERENCE PRocennlNcs Boox, Tth INroRNarroual Colrr,ERENCE oN Appltro Arual,ysrs eNo _.1 *u io-iq-, t01;,"j;r *euL-TuRr(Ey spectrum of cayley graphs of dihedrar groups and their energy Amira Fadina Ahmad Fadzil, Nor Haniza Sarmin and Ahmad Erfanian l:"partment of Mathematical Sci,ences, Faculty of Sci,ence, [Jniuersiti Teknologi, MaLagsia, Johor Bahru, Malaysia. Department of Pttre Mathemat'ics, Faculty of Mithematr,cal Sci,ences and Center oJ Ercellence in Analysis on Algebrazc Structures, Ferdnwli Uniuers,ity of Mashhad,, Mashhad,, Iran E-mail: [email protected], [email protected] and [email protected]

Ab'stract:Let G be a finite gloup and S be a subset of G where ,9 does not include the identity of G and !:11-erse closed' A Cayley graph of 's G with respect. to ,S is a graph where its ve'tices are the elements rl and two vertices o and b are ':i: connecied if ab-1 is in the lubset ,S. 'rhe energy of a Cayley graplr u :he sum of all absolut: ,Y"lY". of the eigenvalues of its adjacency matlix. In this paper., we consider a .pecific subset S : of dihedlal group {o,n/2,b} of order 2n, where rt, } \ and n is even and find t}re ;--a"-ie)' glaph with respect to the set. we also calculate theil spectrurn and compute the energy of :it respected Cayley graphs. Finally, the generalization of the Cayley graphs and their energies a'e found. Keyrvords: Enelgy graph, of eigenvarues, adjacency matrix, cayley graph, Dihedral grorp.

Introduction

The study on the spectrum of Cayley glaphs by using algebraic graph theory was fir'st initiated by Babai li] in 1979 a'd the method has been developed by many researchers. one of the many researchers, c'etkovic has also studied [2] on the spectrurn of graphs. one of ]ris results which is on cayley graphs of dihedral groups gives the following; if all eigenvalues of | : Cay(D2,, ,S) are symmetric with respect to the origin, then the I is bipartite. Although, there are also other approaches to find the spectrum of a graph where Diaconis and Shahshahani [3] have calculated the spectrum of Cayley glaphs by usi'g the chalacter table of the groups.

Besides, the studv on the energy general of simple graphs was first clefined by Gutman in 197g inspired n'orn the Huckel Molecular olbital (HMo) Theory proposed in 1g30s. The Huckel Molecular orbital Theory has been used by chernists in approximating the energies related with n-electr.on orbitals in conjugated hydrocarbon molecules [4]. In the early days, when computers were rrot widely available, tire calculation of the HMo total zr-electron energy was a serious problem, In order to overcome the difficulty, a variety of approaches have been offeled to calculate the approxirnate calculatiorr of the ;i-electron energy' Within the HMO approximation, the total energy of the zr-electrons, denoted by r is obtained by summing distinct electron energies. In conjugated hydrocarbons, the total number of zr-electrons is equal to the number of vertices of the associated molecuiar. glaph. After some considerations, they arrived at the definition of the energy which is the sum of the absolute values of the eigenvalues of the graph [5].

\Iany researchers have studied on the topic of energy of graphs. pati rn 2004, Bapat and [6] have proved that the energy of a graph is never arr ocld integer. Ivleanwhile, the properties that the enelgy of a graph is rrever the square root of an odd integer has been proven by Pirzada and Gutrnan [7] in 200g. There are also a few other researchers who studied specifically on the energy of unitar.y cayley graphs (see [g],

65 coNrpRnNce PRocp'nlrucs.Eoox, 7th INTERNATToTeI coNpERENCE or,r Appllen i, jrro*uu"_Tunxey Aruelysrs arun

Besides' there have also been rnany studies on the caylev graphs for dihedral groups. In 2006, wang arld Xu [10] have considered the non-normal one-regular and 4-valent cayley graphs of dihedral groups while Kwak and oh [11] have classified the 4-valent and 6-varent one regular norrnal cayley graphs of dihedral groups whose vertex stabilizers in Aut(f) are cyclic. In addition, in 200g, Kwak et at.[r2] lrave explored on the one-regular cayley graphs on dihedral groups of any prescribed valency. Kim et al. [13] also have studied on the cayley graphs of dihedral groups on the classification of p-valent regular Cayley graphs.

The objective of this study is to calculate the energy of the cayley graphs associated to dihedral groups for the subset S : {an/2,b} where n ) 3 and n is even, The methodology includes the finding of elements. vertices and edges for the cayley graphs of the dihedral groups, finding their isomorphism, building the adjacency matrix for the cayley graph' finding the spectrum of the adjacency matrix of the graphs and Iastly calculating the energy of the graphs.

2 Preliminaries

Definition 2.L (I4), Dihedral Group rf rn is a regular polygon with n vertices, 'ur,'u2, . . . ,un and center o, then the symmetry group is called the dihedral group D(nn, with 2n elements, and it is denoted by Dzn. Definition 2.2 (Ib). Cayley Graph of a Group Let G be a finite group with identity 1. Let,5 be a subset of G satisfying 1 ( ,s a'd,s:,s-r; thar is' s € if artd 'S only if s-1 e .9. The Cayley graph Cay(G;^9) on G rvith connection set s is clefined a-s follows: o the vertices are the elements of G o tlrere is an edge joining g and, h if and only if h: sg for some s € ,S. The set of all Cayley graphs on G is denoted by Cay(G). Deffnition 2.8 (16). Adjacency Matrix Letf beagraphwithT(f) :{1,.'.'n}andE(f) : {"r,...,e-}. Theadjacencymatrixof f denoted by 'a(f) is the n x n matrix defined as follows. The rows and the columns of ,a(f) are indexed by lz(f;. rf i* thenthe (i,7)-entryof i ,4(l) is 0 forvertices iand nonadjacent, and the (2,7)-entryis jadjacent. i l for f and The(2,z)-entryof ,a(r) is0foli:r,...,n. A(r)isoftensimplydenotedby.4.

Proposition 2']- (17)' cons'id,er the und,irected, n-cycle cn The spectrum conszsts of the numbers {zcos(ff)}; j : {0,r,. . .,n - \} Definition 2.4 (16). Energy of Graph For any graph f, the energy of the g'aph is defined as e(f) : DLr r)ar where ,\r,,..,),, are the eigenvalues of the adjacency matrix of r'

oo CoNnnRsNco PRocnnurNcs Bgox, 7th I*pnN,arrroNAL Corunnnorvce oru AppueD ANALysrs AND M.qlnur,rRrrcAL MoDELTNG ZCa/t"YM'18,) JuNn 20-24, 201g, Isrnrueul-TuRxey 3 Main results

In this section, the main results are specified in term of proposition, lemma and theorem. proposition 2 presents the generalization of the Cayley graphs of D2n with respect to the set S' : {on/, ,b} while Lemma 1 states the spectrum of the Cayley graphs in Pr'<-rposition 2. Then, the results on the energy of the Cayley graphs with respect to the ,S : set {o"/',b} are presented in Theolem 2. The findings in proposition 2 and Lemma I will be used in proving Theorern 2. Oneexample is also presented for better understandinss.

Proposition 3.1. Let D2n be a dihedral group of ord,er2n wheren) 3 and.n enen, Let s : {ar/2,b} beasubsetof Dzn.TheCayleygraphsof Dznwr,th.respecttothesets,Cay(D2n,{an/z,b}),arethecyclLe sraphs Uf Cn.

Considel Proof the dihedral groups D2n of ord,er 2n and the Cayley graphs of D2n with respect to the subset S : {an/2,b}, denoted as Cay(D2n,{on/r,b}). Note that lSl :2. By the clefinition of Cayley graph, the veltices of Cag(D2n,{on/',b}) are the elements ofD2n and there is an edge joining vertices h and g if and only if hq-r e ,5. Since the order of ,S is 2, this forms f cycle graphs of order 4 where all vertices are from the group Drr,l_)P Cn. I Lemma 3.1. Let D2n be a dtlrcdral group of orcler2n wheren> 3 and,n euen. Let S: {on/r,b} be a subset of Dzn. Thus, th.e spectrunt of the cay(D2n,{an/2,b}), d,enoted, as spec(caa(Dzn,{on/r,b})) are, 0 with muLti,plicity n and *2 wi,th muttiplicity to nl2. Proof Consider the dihedral gloups D2,, of order 2n. By Proposition 2, The Cayley graphs of D2,, with respect to the subset S , Cay(D2n, {an/z,b}), ale the cycle graph lJf Cn Since the adjacency spectrum of a cycle g'aph C,, is : ..,n 1}, {2cos(ff)};,i {0,1,. - rhen Sztece! Ci : }{zcos(!)};l : {0, I,2,8} n'lrich also can be written as ) : 0 with multiplicity n. and .\ : i2 with multiplicity to nl2. I

Theorem 3.1. Let D2n be a dthed'ral group of order2n uherent S and, S: {an/2,b} i,s a subset of 'lh'e D2n. energ!) of the cayley graphs of Dzn w'ith respect to the set s, E(cuy(D2r,{an/2,b})):zn. Proof Consider the dihedral groups D2n of. order 2n. By Proposition 2 and Lemma 1, The Cayley graphs of D2,, with respect to the subset s , cay(D2r,{an/2,b}), are the cycle graph tJl cn with t}re spectrum :0 ) with muitiplicity n and \: !2 with multiplicity to nf2. Tlter-efore, the energy of the Cayley graphs of D2n with respect to the subset ,9, E(Cay(D2", {o,/,,b})) : nlOl + tlzl + El - zl : 2". I The computation of the energy of Cayley graph respect '*,ith to the set S : {o, , b} of dihecilal groups of order 8, Ds is as shown in the following example.

Exarrrple 3.1. Let Ds be a dihedr.al group of order B where Da :< a,blaa :b2 : !,bab: o-1 > and S:{a2,b}beasubsetof Ds. T}reCayleygraphof Dswithrespecttothesubset S,Cay(Ds,{a2,b}) a.e tlre cycle graph lJ?cn.'Ihe spectrum of Cay(Da,{a2,b}) are 0 with multiplicity 4 and *2 with multi- plicity 2. Thus, the enelgy of the Cayley glaphs of Ds with respect to the subset S, E(Cay(Da, {a2, b1;; is 8.

o/ CoNrennNcp PRocpporr.rcs Boox, 7th INTERNATToTnI CoNT,ERENcE oN Appr,rpo Aunlysrs AND MetuptraRucnl, Moonr,ruc LCAAMMI8,) Jurun 20-24, 2018, IsmNsur,-TuRxey

Proof Consider a dihedral group of order De :< a,blaa : b2 : I,bab: a-L > and ,S : {a2,b} be a subset of D3. Then, by Definition 2, the vertex r is connQcted to sr where s € ,9.

I - a2 since a2.I : a2, b - a2b since a2.b : a2b, 1-bsinceb.I:b. b-lsinceb.b:1, a - a3 since a2.a : a3. ab - alb since a2,ab - a3b, a - a3b since b.a : asb, ab - a3 since b.ab : a3-. a2-!sincea2.a2:r, azb-bsincea2.a2b:b, a2 - a2b since b,a2 :0,2b, azb - a2 since b.a2b : i2. a3-asincea2.o3:a. a3b - ab since a2.a3b : ab. a3 - ab since b.a3 : ab. a3b-asinceb.a3b:a.

The connected elements form the cay(Ds,{o',b}):gl,ca as illustrated in Figure 1.

Figure L: C ay(Ds, {o', b})

By the definition of adjacency matrix, 00101000 00010001 10000010 01000100 A(Cay(D6, : {a2,U111 10000010 00010001 00101000 01000100 FYom the adjacency matrix, the characteristic polynomials found is /(f) : A8 - g)6 + 16A4 which gives the spectrum, ,\ : 0 with multiplicity 4 and ) : *2 with multiplicity 2. By using the generalization of spectrum of a cycle graph, the spectrum can also be found as Spec(J?o Ca) : 2{2cos(f;),2gos(f,),2cos(ff),Zcos(ff)) :2{2,0, -2,0} which.can be written as ) : 0 wirh multiplicity 4 and .\ : t2 with multiplicity 2. Therefore, E(Cay(Ds, {a2,b})): 4l0l + 2l2l + 2l - 2l:8. I

68 CoNrpRpr.rcp PRoceoorr.rcs Bgoxr Tth lrurpnN,qrroNel coNpDRENCE oN Appr,r'o AN.^rvsrs Rwo Mersevntlcal MoonlrNc ICAAMM|8,) Jurue 20-24, 201g, Isrnueul_Tunxsv 4 Conclusion

As a conclusion, it has been found that the energy of the Cayley graphs of the dihedral groups of order 2n with respect to the subset S : {an/2,b} where n ) 4 and n is even at-e 2n.

Acknowledgment

The authors would like to expless their appreciation for the support of the sponsor; Ministry of Higher Education (MOHE) N{alaysia and Research Management Centle (RMC), Universiti Teknologi Malaysia (UTM) Johor Bahru for the financial funding through the Research University Grant (GUp) Vote No. 13H79 and 13J82. The first author wor,rld also like to express her appreciation to UTM for the financial support under Zamalah Scholalshio.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authols have contributed to all parts ofthe article. AII authors read and approved the final manuscript.

References

Babai, (1979). [1] L' Spectra of Cayley graphs. Journal of Combinatorial Theory, Series B,2T(2), Ig0- 189.

Cvetkovic, [2] D. M., Rowlinson, P., and Simic, S. (2010). An introduction to the theory of graph spectra, Cambridge, UK: Cambridge University press.

Diaconis, P., and [3] Shahshahani, i\4. (i98i). Generating a random permutation with random transpo- sitions' Zeitschrift fiir Wahrscheinlichkeitstheorie und verwandte Gebiete , bT(2),159-179. (2013). [4] Woods, C. My Favorite Application Using Graph Eigenvalues: Glaph Energy.

[5] Li' X., Shi., and Gutman, I. (2012). Graph Energy. Springer Science and Business Media. Bapat, [6] R'' B', and Pati, S. (2004). Energy of a graph is never an odd integer. Bull. Kerala Math. Assoc, I(2),129-132.

Pirzada, S., and [7] Gutman, I. (2008). Energy of a graph is never the square root of an odd integer. Applicable Analysis and Discrete Mathematics, 2(1), 11g_121.

Foster-Greenwood, [B] B. and Kriloff, C. Spectra of Cayley graphs of complex reflection g1oups. Journal of Algebraic Combinatorics 44(1) (2016), 33-52.

69 CoNpeRpNcs PRocneotNcs ,Eoox, 7th Inrenxerrome,l ColrpERENcE oN Appllso Au.4rysrs nnn Mnrnsrunrrcnr, MonnllNc (zc,4l,AzIl1g,) Juruo zo-iq,, iors,-I.ro"euL_TuRxey Liu, X' and Li, B. Distance [9] powers of unitary Cayley graphs, Applied Mathematics and Computation 289 (2016), 272-280.

Wang, C' and Xu' M., Non-normal [10] one-regular and 4-valent Cayley graphs of dihedral groups D2rr. European Journal of Combinatorics, 27(b), pp.Tb0_266, 2006.

Kwak, J.H. and Oh, [11] J.M., OneUregular Normal Cayley Graphs on Dihedral Groups of Valency 4 or 6 with Cyclic Vertex Stabilizer. Acta Mathematica Sinica, 22(S), pp.130b-1320, 2006. Kwak, [iz] J.H', Kwon, Y.S' and Oh, J.M,, Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency. Journal of Combinatorial Theory, Series B, g8(3), pp.bgb-bgg, 200g.

Kim, D. S., Kwon, [13] Y.S', and Lee, J, U. (2010). A classification of prime-valent regular Cayley maps on abelian, dihedral and dicyclic groups. Bulletin of the Korean Mathematical Society, 4T (l), 17-27 .

[14] Rotman, J. J., Advanced Modern Algebla, usA: pearson Education,2002.

Beineke, [is] L. w. and wilson, R. J., Topics in algebraic graph theory, 102, usA : Cambridge University Press. 2004.

[16] Bapat, R, B., Graphs and matrices, New yoLk (Ny): Springer, 2010.

Brouwer, [17] A.E. and Haemers. w.H., spectra of graphs. springer science & Business Media. (2011).

"{ .) s

'.i

t i 'I

n

lr

70