Eigenvalues of Cayley graphs Xiaogang Liu Department of Applied Mathematics Northwestern Polytechnical University Xi’an, Shaanxi 710072, P. R. China [email protected] Sanming Zhou School of Mathematics and Statistics The University of Melbourne Parkville, VIC 3010, Australia [email protected] January 22, 2019 Abstract We survey some of the known results on eigenvalues of Cayley graphs and their applica- tions, together with related results on eigenvalues of Cayley digraphs and generalizations of Cayley graphs. Keywords: Spectrum of a graph; Eigenvalues of a graph; Cayley graphs; Integral graphs; Energy of a graph; Ramanujan graph; Second largest eigenvalue of a graph; Distance-regular graphs; Strongly regular graphs; Perfect state transfer AMS Subject Classification (2010): 05C50, 05C25 Contents 1 Introduction 4 1.1 Outlineofthepaper ............................... ... 5 1.2 Terminologyandnotation . ..... 7 arXiv:1809.09829v2 [math.CO] 22 Jan 2019 2 Eigenvalues of Cayley graphs 9 2.1 Characters...................................... 9 2.2 EigenvaluesofCayleygraphs . ...... 10 2.3 Eigenvalues of vertex-transitive graphs . ............ 12 3 Integral Cayley graphs 13 3.1 Characterizations of integral Cayley graphs . ............ 13 3.1.1 Integral circulant graphs, unitary Cayley graphs, and gcd graphs . 13 3.1.2 Integral Cayley graphs on abelian groups . ....... 15 1 3.1.3 IntegralnormalCayleygraphs . 16 3.1.4 Integral Cayley multigraphs . ..... 16 3.2 A few families of integral Cayley graphs on abelian groups............. 18 3.2.1 Unitary finite Euclidean graphs . ..... 18 3.2.2 NEPS of complete graphs, gcd graphs of abelian groups, and generalized Hamminggraphs................................ 19 3.2.3 Sudoku graphs and positional Sudoku graphs . ....... 20 3.2.4 Pandiagonal Latin square graphs . ..... 21 3.3 Integral Cayley graphs on non-abelian groups . ........... 21 3.3.1 Cayleygraphsondihedralgroups. ..... 21 3.3.2 Cayley graphs on dicyclic groups . ..... 23 3.3.3 Cayley graphs on a family of groups with order 6n ............. 24 3.3.4 Cayleygraphsonsymmetricgroups . 24 3.4 Integral Cayley graphs of small degrees . .......... 27 3.5 Automorphism groups of integral Cayley graphs . ........... 28 3.5.1 Cayleyintegralgroups . 28 3.5.2 Cayley integral simple groups . ..... 29 3.5.3 Automorphism groups of integral circulant graphs . .......... 30 3.6 Cayley digraphs integral over the Gauss field or other numberfields . .. .. 31 4 Cospectral Cayley graphs 32 4.1 Cospectral Cayley graphs on non-abelian groups . ........... 32 4.2 Cospectralityandisomorphism . ....... 33 5 Cayley graphs on finite commutative rings 35 5.1 Unitary Cayley graphs of finite commutative rings . ........... 36 5.2 Quadratic unitary Cayley graphs of finite commutative rings ........... 37 5.3 A family of Cayley graphs on finite chain rings . ......... 39 6 Energies of Cayley graphs 39 6.1 Energies of integral circulant graphs . .......... 40 6.2 Extremal energies of integral circulant graphs . ............. 45 6.3 Energiesofcirculantgraphs . ....... 47 6.4 Energies of unitary Cayley graphs and quadratic unitary Cayley graphs of finite commutativerings ................................... 48 6.5 Energies of Cayley graphs on finite chain rings and gcd graphs of unique factor- izationdomains..................................... 50 6.6 Skew energy of orientations of hypercubes . .......... 51 2 6.7 Others.......................................... 51 7 Ramanujan Cayley graphs 52 7.1 Ramanujan integral circulant graphs . ......... 52 7.2 Ramanujan Cayley graphs on dihedral groups . ......... 55 7.3 Ramanujan Cayley graphs on finite commutative rings . ........... 55 7.4 RamanujanEuclideangraphs . ..... 57 7.5 Ramanujanfiniteupperhalfplanegraphs . ........ 59 7.6 Heisenberggraphs ................................ 61 7.7 Platonicgraphsandbeyond . ..... 61 7.8 A family of Ramanujan Cayley graphs on Zp[i] ................... 62 7.9 Bounds for the degrees of Ramanujan Cayley graphs . .......... 62 8 Second largest eigenvalue of Cayley graphs 64 8.1 CayleygraphsonCoxetergroups . ...... 64 8.2 Normal Cayley graphs on highly transitive groups . ........... 66 8.3 Three Cayley graphs on alternating groups . ......... 66 8.4 Cayleygraphsonabeliangroups . ...... 67 8.5 A family of Cayley graphs with small second largest eigenvalues . 67 8.6 FirsttypeFrobeniusgraphs . ...... 68 8.7 Distance-j Hamming graphs and distance-j Johnsongraphs . 68 8.8 Algebraicallydefinedgraphs. ....... 69 9 Perfect state transfer in Cayley graphs 71 9.1 Perfect state transfer in Cayley graphs on abelian groups ............. 72 9.2 A few families of Cayley graphs on abelian groups admitting perfect state transfer 73 9.2.1 Perfect state transfer in integral circulant graphs . ............. 73 9.2.2 Perfect state transfer in cubelike graphs . ......... 73 9.2.3 Perfect state transfer in gcd graphs of abelian groups ............ 75 9.3 Perfect state transfer in Cayley graphs on finite commutative (chain) rings and gcd graphs of unique factorization domains . ....... 76 10 Distance-regular Cayley graphs 77 10.1 Distance-regular Cayley graphs . ......... 78 10.2 Distance-regular Cayley graphs with least eigenvalues 2 ............. 81 − 11 Generalizations of Cayley graphs 82 11.1 Eigenvalues of n-Cayleygraphs ............................ 82 11.2 Strongly regular n-Cayleygraphs ........................... 84 3 11.3 Cayleysumgraphs ................................ 85 11.4 Group-subgrouppairgraphs. ....... 87 12 Directed Cayley graphs 87 12.1 Eigenvalues of directed Cayley graphs . .......... 87 12.2 Two families of directed Cayley graphs . .......... 89 12.3 Two more families of directed Cayley graphs . .......... 89 13 Miscellaneous 91 13.1 RandomCayleygraphs . .. .. .. .. .. .. .. .. 91 13.2 Distance eigenvalues of Cayley graphs . .......... 92 13.3Others......................................... 93 1 Introduction The study of eigenvalues of graphs is an important part of modern graph theory. In particular, eigenvalues of Cayley graphs have attracted increasing attention due to their prominent roles in algebraic graph theory and applications in many areas such as expanders, chemical graph theory, quantum computing, etc. A large number of results on spectra of Cayley graphs have been produced over the last more than four decades. This paper is a survey of the literature on eigenvalues of Cayley graphs and their applications. All definitions below are standard and can be found in, for example, [59, 92, 133]. A finite undirected graph consists of a finite set whose elements are called the vertices and a collection of unordered pairs of (not necessarily distinct) vertices each called an edge. As usual, for a graph G, we use V (G) and E(G) to denote its vertex and edge sets, respectively, and we call the size of V (G) the order of G. An edge u, v of G is usually denoted by uv or vu. If uv is an edge { } of G, we say that u and v are joined by this edge, u and v are adjacent in G, and both u and v are incident to the edge uv. An edge joining a vertex to itself is called a loop, and two or more edges joining the same pair of distinct vertices are called parallel edges. The degree of u in G, denoted by dG(u) or simply d(u) if there is no risk of confusion, is the number of edges of G incident to u, each loop counting twice. A graph without loops is called a multigraph, and a graph is simple if it has no loops or parallel edges. Let G be a finite undirected graph of order n. The adjacency matrix of G, denoted by A(G), is the n n matrix with rows and columns indexed by the vertices of G such that the (u, v)-entry × is equal to the number of edges joining u and v, with each loop counting as two edges. The eigenvalues of A(G) are called the eigenvalues of G, and the collection of eigenvalues of G with multiplicities is called the spectrum of G. Since G is undirected, A(G) is a real symmetric matrix and hence the eigenvalues of G are all real numbers. If λ1, λ2, . , λr are distinct eigenvalues of G and m1,m2,...,mr the corresponding multiplicities, then the spectrum of G is denoted by λ λ . λ Spec(G)= 1 2 r m m ... m 1 2 r or m1 m2 mr (λ1 , λ2 , . , λr ), 4 where in the latter notation we usually omit mi if mi = 1 for some i. An eigenvalue with multiplicity 1 is called a simple eigenvalue. The spectral radius of a graph is the maximum modulus of its eigenvalues. Two graphs are said to be cospectral if they have the same spectrum. Let D(G) be the n n diagonal matrix whose (u, u)-entry is equal to the degree d(u) of u in × G, for each u V (G). The matrices D(G) A(G) and D(G)+ A(G) are called the Laplacian ∈ − matrix and signless Laplacian matrix of G, respectively, and their eigenvalues are called the Laplacian eigenvalues and signless Laplacian eigenvalues of G, respectively. In the case when G is k-regular for some integer k 0, that is, each vertex has degree k, a real number x is an ≥ eigenvalue of G if and only if k x is a Laplacian eigenvalue of G. Since all graphs considered in − this paper are regular, this implies that all results to be reviewed in the paper can be presented in terms of Laplacian eigenvalues. Henceforth we will mostly talk about eigenvalues of graphs. Similar to undirected graphs, a finite directed graph (or digraph) can be defined by specifying a finite set of vertices and a collection of ordered pairs of (not necessarily distinct) vertices; each ordered pair (u, v) in the collection is called an arc from u to v. The underlying graph of a digraph is the undirected graph obtained by replacing each arc by an edge with the same end- vertices. The adjacency matrix of a finite digraph G is the matrix whose (u, v)-entry is equal to the number of arcs from u to v, and the eigenvalues of this matrix are called the eigenvalues of G. Note that, unlike the undirected case, a digraph may have complex eigenvalues as its adjacency matrix is not necessarily symmetric. A digraph without loops is called a multidigraph.