Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 289–299 GCD-Graphs and NEPS of Complete Graphs Walter Klotz Institut fur¨ Mathematik, Technische Universitat¨ Clausthal, Germany Torsten Sander Fakultat¨ fur¨ Informatik, Ostfalia Hochschule fur¨ angewandte Wissenschaften, Germany Received 21 February 2012, accepted 20 August 2012, published online 19 November 2012 Abstract A gcd-graph is a Cayley graph over a finite abelian group defined by greatest common divisors. Such graphs are known to have integral spectrum. A non-complete extended p- sum, or NEPS in short, is well-known general graph product. We show that the class of gcd-graphs and the class of NEPS of complete graphs coincide. Thus, a relation between the algebraically defined Cayley graphs and the combinatorially defined NEPS of complete graphs is established. We use this link to show that gcd-graphs have a particularly simple eigenspace structure, to be precise, that every eigenspace of the adjacency matrix of a gcd- graph has a basis with entries −1; 0; 1 only. Keywords: Integral graphs, Cayley graphs, graph products. Math. Subj. Class.: 05C25, 05C50 1 Introduction n Given a set B ⊆ f0; 1g and graphs G1;:::;Gn, the NEPS (non-complete extended p- sum) of these graphs with respect to basis B, G = NEPS(G1;:::;Gn; B), has as its vertex set the Cartesian product of the vertex sets of the individual graphs, V (G) = V (G1)×· · ·× V (Gn). Distinct vertices x = (x1; : : : ; xn); y = (y1; : : : ; yn) 2 V (G) are adjacent in G, if and only if there exists some n-tuple (β1; : : : ; βn) 2 B such that xi = yi, whenever βi = 0, and xi; yi are distinct and adjacent in Gi, whenever βi = 1. In particular, NEPS(G1; f(1)g) = G1 and NEPS(G1; ;) = NEPS(G1; f(0)g) is the graph without edges on the vertices of G1. The NEPS operation generalizes a number of known graph products, all of which have in common that the vertex set of the resulting graph is the Cartesian product of the input vertex sets. For example, NEPS(G1;:::;Gn; f(1; 1;:::; 1)g) = G1 ⊗ ::: ⊗ Gn is the E-mail addresses: [email protected] (Walter Klotz), [email protected] (Torsten Sander) Copyright c 2013 DMFA Slovenije 290 Ars Math. Contemp. 6 (2013) 289–299 product of G1;:::;Gn (cf. [10], “direct product” in [15]). As can be seen, unfortunately, the naming of graph products is not standardized at all. The “Cartesian product” of graphs in [15] is even known as the “sum” of graphs in [10]. With respect to this seemingly ar- bitrary mixing of sum and product terminology, let us point out that here the term “sum” (and also the “p-sum” contained in the NEPS acronym) indicates that the adjacency matrix of the constructed product graph arises from a certain sum of matrices (involving the adja- cency matrices of the input graphs). Refer to [10] or [11] for the history of the notion of NEPS. We remark that the NEPS operation can be generalized even further, see e.g. [12] and [21]. Next, we consider the important class of Cayley graphs [13]. These graphs have been and still are studied intensively because of their symmetry properties and their connections to communication networks, quantum physics and other areas [8], [13]. Let Γ be a finite, additive group. A subset S ⊆ Γ is called a symbol (also: connection set, shift set) of Γ if −S = {−s : s 2 Sg = S, 0 62 S. The undirected Cayley graph over Γ with symbol S, denoted by Cay(Γ;S), has vertex set Γ; two vertices a; b 2 Γ are adjacent if and only if a − b 2 S. Let us now construct the class of gcd-graphs. The greatest common divisor of non- negative integers a and b is denoted by gcd(a; b), gcd(0; b) = gcd(b; 0) = b. If x = (x1; : : : ; xr) and m = (m1; :::; mr) are tuples of nonnegative integers, then we set gcd(x; m) = (d1; : : : ; dr) = d; di = gcd(xi; mi) for i = 1; : : : ; r: For an integer n ≥ 1 we denote by Zn the additive group of integers modulo n, the ring of integers modulo n, or simply the set f0; 1; : : : ; n − 1g. The particular choice will be clear from the context. Let Γ be an (additive) finite abelian group represented as a direct sum of cyclic groups, Γ = Zm1 ⊕ ::: ⊕ Zmr ; mi ≥ 1 for i = 1; : : : ; r: Suppose that di is a divisor of mi; 1 ≤ di ≤ mi, for i = 1; : : : ; r. For the divisor tuple d = (d1; : : : ; dr) of m = (m1; : : : ; mr) we define SΓ(d) = fx = (x1; : : : ; xr) 2 Γ : gcd(x; m) = dg: Let D = fd(1); : : : ; d(k)g be a set of distinct divisor tuples of m and define k [ (j) SΓ(D) = SΓ(d ): j=1 Observe that the union is actually disjoint. The sets SΓ(D) shall be called gcd-sets of Γ. We define the class of gcd-graphs as the Cayley graphs Cay(Γ;S) over a finite abelian group Γ with symbol S a gcd-set of Γ. The most prominent members of this class are perhaps the unitary Cayley graphs Xn = Cay(Zn;Un), where Un = SZn (1) is the multiplicative group of units of Zn (cf. [16], [17], [22]). The main goal of this paper is to show in Section2 that every gcd-graph is isomorphic to a NEPS of complete graphs. Conversely, every NEPS of complete graphs is isomorphic to a gcd-graph over some abelian group. This relation is remarkable since it allows us to define gcd-graphs either algebraically (via Cayley graphs) or purely combinatorially (via NEPS). The characterization of gcd-graphs as NEPS of complete graphs reveals some new access to structural properties of gcd-graphs. As a first application, we show in Section W. Klotz and T. Sander : GCD-Graphs and NEPS of Complete Graphs 291 3 that every gcd-graph has simply structured eigenspace bases for all of its eigenvalues. This means that for every eigenspace a basis can be found whose vectors only have entries from the set f0; 1; −1g. It is known that other graph classes exhibit a similar eigenspace structure, although not necessarily for all of their eigenspaces [9], [20], [25]. Finally, we present some open problems in Section4. 2 Isomorphisms between NEPS of complete graphs and gcd-graphs We are going to show in several steps that gcd-graphs and NEPS of complete graphs are the same. Lemma 2.1. Let Γ = Zm1 ⊕ · · · ⊕ Zmr and d = (d1; : : : ; dr) a tuple of positive divisors r of m = (m1; : : : ; mr). Define b = (bi) 2 f0; 1g by ( 1 if di < mi; bi = 0 if di = mi: Then we have Cay(Γ;S (d)) = NEPS(Cay(Z ;S (d ));:::; Cay(Z ;S (d )); fbg): Γ m1 Zm1 1 mr Zmr r Proof. Both Cay(Γ;SΓ(d)) and the above NEPS have the same vertex set Γ. It remains to show that they have the same edge set. Let x; y 2 Γ with x = (x1; : : : ; xr), y = (y1; : : : ; yr) and suppose that x 6= y. Now x and y are adjacent in Cay(Γ;SΓ(d)) if and only if gcd(xi − yi; mi) = di for i = 1; : : : ; r. The latter condition means that in case di < mi the vertices xi and yi are adjacent in G = Cay(Z ;S (d )), and in case d = m we have x = y . But this is exactly the i mi Zmi i i i i i condition for adjacency of x and y in NEPS(G1;:::;Gr; fbg). The following lemma allows us to break down the Cayley graphs that form the factors of the NEPS mentioned in Lemma 2.1. Each factor can be transformed into a gcd-graph over a product of cyclic groups of prime power order. Using Lemma 2.1 once again, we obtain a representation of the original graph as a NEPS of NEPS of gcd-graphs over cyclic groups of prime power order. Lemma 2.2. Let the integer m ≥ 2 and a proper divisor d ≥ 1 of m be given as products of powers of distinct primes, r Y αi m = mi; mi = pi ; αi > 0 for i = 1; : : : ; r; i=1 r Y βi d = di; di = pi ; 0 ≤ βi ≤ αi for i = 1; : : : ; r: i=1 ~ If we set Γ = Zm1 ⊕ · · · ⊕ Zmr and d = (d1; : : : ; dr), then there exists an isomorphism ~ Cay(Zm;SZm (d)) ' Cay(Γ;SΓ(d)): 292 Ars Math. Contemp. 6 (2013) 289–299 Proof. By the Chinese remainder theorem [23] we know that every z 2 Zm is uniquely determined by the congruences z ≡ zi mod mi; zi 2 Zmi for i = 1; : : : ; r: This gives rise to a bijection Zm ! Γ by virtue of z 7! (z1; : : : ; zr) =:z ~. We show that ~ this bijection induces an isomorphism between Cay(Zm;SZm (d)) and Cay(Γ;SΓ(d)). ~ Let x; y 2 Zm, x 6= y. Note that x~ and y~ are vertices of Cay(Γ;SΓ(d)). The vertices x and y are adjacent in Cay(Zm;SZm (d)) if and only if gcd(x−y; m) = d. This is equivalent ~ to gcd(xi − yi; mi) = di for every i = 1; : : : ; r. Now this means gcd(~x − y;~ m~ ) = d, with ~ m~ = (m1; : : : ; mr), which is the condition for adjacency of x~ and y~ in Cay(Γ;SΓ(d)). Next we shall prove a lemma that helps us consolidate the nesting of NEPS operations into a single NEPS operation.