On 4-semiregular 1-factorizations of complete graphs and complete bipartite graphs

A 4-semiregular 1-factorization is a 1-factorization in which every pair of distinct 1-factors forms a union of 4-cycles. LetK be the complete graphK _2nor the complete bipartite graphK _{n, n} .We prove that there is a 4-semiregular 1-factorization ofK if and only ifn is a power of 2 andn2, and 4-semiregular 1-factorizations ofK are isomorphic, and then we determine the symmetry groups. They are known for the case of the complete graphK _2n ,however, we prove them in a different method.     

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.     

Transformation graph

The transformation graph G^-+- of a graph G is the graph with vertex set V(G)E(G), in which two vertices u and v are joined by an edge if one of the following conditions holds: (i) u,vV(G) and they are not adjacent in G, (ii) u,vE(G) and they are adjacent in G, (iii) one of u and v is in V(G) while the other is in E(G), and they are not incident in G. In this paper, for any graph G, we determine the connectivity and the independence number of G^-+-. Furthermore, for a graph G of order n4, we show that G^-+- is hamiltonian if and only if G is not isomorphic to any graph in {2K₁+K₂,K₁+K₃}{K_1,n-1,K_1,n-1+e,K_1,n-2+K₁}.     

The energy of unitary cayley graphs

A graph G of order n is called hyperenergetic if E(G)>2n-2, where E(G) denotes the energy of G. The unitary Cayley graph X_n has vertex set Z_n={0,1,2,…,n-1} and vertices a and b are adjacent, if gcd(a-b,n)=1. These graphs have integral spectrum and play an important role in modeling quantum spin networks supporting the perfect state transfer. We show that the unitary Cayley graph X_n is hyperenergetic if and only if n has at least two prime factors greater than 2 or at least three distinct prime factors. In addition, we calculate the energy of the complement of unitary Cayley graph and prove that is hyperenergetic if and only if n has at least two distinct prime factors and n≠2p, where p is a prime number. By extending this approach, for every fixed , we construct families of k hyperenergetic non-cospectral integral circulant n-vertex graphs with equal energy.     

Strongly Regular Semi-Cayley Graphs

We consider strongly regular graphs = (V, E) on an even number, say 2n, of vertices which admit an automorphism group G of order n which has two orbits on V. Such graphs will be called strongly regular semi-Cayley graphs. For instance, the Petersen graph, the Hoffman–Singleton graph, and the triangular graphs T(q) with q 5 mod 8 provide examples which cannot be obtained as Cayley graphs. We give a representation of strongly regular semi-Cayley graphs in terms of suitable triples of elements in the group ring Z G. By applying characters of G, this approach allows us to obtain interesting nonexistence results if G is Abelian, in particular, if G is cyclic. For instance, if G is cyclic and n is odd, then all examples must have parameters of the form 2n = 4s ² + 4s + 2, k = 2s ² + s, = s ² – 1, and = s ²; examples are known only for s = 1, 2, and 4 (together with a noncyclic example for s = 3). We also apply our results to obtain new conditions for the existence of strongly regular Cayley graphs on an even number of vertices when the underlying group H has an Abelian normal subgroup of index 2. In particular, we show the nonexistence of nontrivial strongly regular Cayley graphs over dihedral and generalized quaternion groups, as well as over two series of non-Abelian 2-groups. Up to now these have been the only general nonexistence results for strongly regular Cayley graphs over non-Abelian groups; only the first of these cases was previously known.     

Energy of unitary Cayley graphs and gcd-graphs

This work is based on ideas of Ili? [A. Ili?, The energy of unitary Cayley graphs, Linear Algebra Appl. 431 (2009) 1881-1889] on the energy of unitary Cayley graph. For a finite commutative ring R with unity , the unitary Cayley graph of R is the Cayley graph whose vertex set is R and the edge set is {{a,b}:a,b∈Randa-b∈R^×}, where R^× is the group of units of R. We study the eigenvalues of the unitary Cayley graph of a finite commutative ring and some gcd-graphs and compute their energy. Moreover, we obtain the energy for the complement of unitary Cayley graphs.     

On automorphism groups of Cayley graphs

LetX _G,H denote the Cayley graph of a finite groupG with respect to a subsetH. It is well-known that its automorphism groupA(X_G,H) must contain the regular subgroupL _G corresponding to the set of left multiplications by elements ofG. This paper is concerned with minimizing the index [A(X_G,H)L_G] for givenG, in particular when this index is always greater than 1. IfG is abelian but not one of seven exceptional groups, then a Cayley graph ofG exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.     

Preservation and distortion of area in finitely presented groups

IfK=G where is a tame automorphism of the 1-relator groupG, then the combinatorial area of loops in a Cayley graph ofG is undistorted in a Cayley graph ofK. Examples of distortion of area in fibres of fibrations over the circle are given and a notion of exponent of area distortion is introduced and studied. The inclusion of a finitely generated abelian subgroup in the fundamental group of a compact 3-manifold does not distort area.Partially supported by NSF grant DMS-9200433.     

Largest cliques in connected supermagic graphs

A graph G=(V,E) is said to be magic if there exists an integer labeling f:VE[1,|VE|] such that f(x)+f(y)+f(xy) is constant for all edges xyE.Enomoto, Masuda and Nakamigawa proved that there are magic graphs of order at most 3n²+o(n²) which contain a complete graph of order n. Bounds on Sidon sets show that the order of such a graph is at least n²+o(n²). We close the gap between those two bounds by showing that, for any given connected graph H of order n, there is a connected magic graph G of order n²+o(n²) containing H as an induced subgraph. Moreover G admits a supermagic labeling f, which satisfies the additional condition f(V)=[1,|V|].     

Independent perfect domination sets in Cayley graphs

In this paper, we show that a Cayley graph for an abelian group has an independent perfect domination set if and only if it is a covering graph of a complete graph. As an application, we show that the hypercube Q_n has an independent perfect domination set if and only if Q_n is a regular covering of the complete graph K_n+1 if and only if n = 2^m ? 1 for some natural number m. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 213–219, 2001     

Counting Sets With Small Sumset, And The Clique Number Of Random Cayley Graphs

Given a set A /N we may form a Cayley sum graph G _A on vertex set /N by joining i to j if and only if i+j A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of G _A is almost surely O(logN). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this result we must study the specific structure of set addition on /N. Indeed, we also show that the clique number of a random Cayley sum graph on =(/2) ⁿ is almost surely not O(log ||).* Supported by a grant from the Engineering and Physical Sciences Research Council of the UK and a Fellowship of Trinity College Cambridge.     

On the Classification of Arc-transitive Circulant Digraphs of Order Odd-Prime-Squared

A Cayley graph F = Cay（G, S） of a group G with respect to S is called a circulant digraph of order pk if G is a cyclic group of the same order. Investigated in this paper are the normality conditions for arc-transitive circulant （di）graphs of order p^2 and the classification of all such graphs. It is proved that any connected arc-transitive circulant digraph of order p^2 is, up to a graph isomorphism, either Kp2, G（p^2,r）, or G（p,r）[pK1], where r｜p- 1.     

On the Normality of Cayley Digraphs of Valency 2 on Non-Abelian Group of Odd Order

In this paper, we prove that the Cayley digraph = Cay(G, S) of valency 2 on non-abelian group G of odd order is normal if the automorphism group of A(), a graph constructed from by using the method presented in the paper, is primitive on the vertices set V(A(). We also prove that the Cayley digraphs of valency 2 on non-abelian group of order pq² are normal, where p and q are distinct odd primes.AMS Subject Classification (2000) 05C25 20B25Supported by the National Natural Science Foundation of China (Grant no. 19971086) and the Doctoral Program Foundation of the National Education Department of China.     

Some graphs with small second eigenvalue

For any prime,p, we construct a Cayley graph on the group,G, of affine linear transformations ofℤ/pℤ of degree 2(p−1) and second eigenvalue with the following special property: the adjacency matrix of the graph is supported on the “blocks” associated to the trivial representation and the irreducible representation of sizep−1. SinceG is of orderp(p−1), the correspondingt-uniform Cayley hypergraph has essentially optimal second eigenvalue for this degree and size of the graph (see [2] for definitions). En route we give, for any integerk>1, a simple Cayley graph onp ^k nodes of degreep of second eigenvalue . The author wishes to acknowledge the National Science Foundation for supporting this research in part under Grant CCR-8858788, and the Office of Naval Research under Grant N00014-87-K-0467.     

Normal Cayley graphs of finite groups

LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G, S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ◃A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S) is normal. It is proved that every finite group G has a normal Cayley graph unlessG≅ℤ₄×ℤ₂ orG≅Q ₈×ℤ ₂ ^r (r⩾0) and that every finite group has a normal Cayley digraph, where Z_m is the cyclic group of orderm and Q₈ is the quaternion group of order 8. Project supported by the National Natural Science Foundation of China (Grant No. 10231060) and the Doctorial Program Foundation of Institutions of Higher Education of China.     

On the average genus of the random graph

The generalized Petersen graph GP (n, k), n ≤ 3, 1 ≥ k < n/2 is a cubic graph with vertex-set {u_j; i ? Z_n} ∪ {v_j; i ? Z_n}, and edge-set {u_iu_i, u_iv_i, v_iv_{i+k, i?}Z_n}. In the paper we prove that (i) GP(n, k) is a Cayley graph if and only if k² ? 1 (mod n); and (ii) GP(n, k) is a vertex-transitive graph that is not a Cayley graph if and only if k² ? -1 (mod n) or (n, k) = (10, 2), the exceptional graph being isomorphic to the 1-skeleton of the dodecahedon. The proof of (i) is based on the classification of orientable regular embeddings of the n-dipole, the graph consisting of two vertices and n parallel edges, while (ii) follows immediately from (i) and a result of R. Frucht, J.E. Graver, and M.E. Watkins [“The Groups of the Generalized Petersen Graphs,” Proceedings of the Cambridge Philosophical Society, Vol. 70 (1971), pp. 211-218]. © 1995 John Wiley & Sons, Inc.     

On the hyperbolicity constant in graphs

If X is a geodesic metric space and x₁,x₂,x₃∈X, a geodesic triangleT={x₁,x₂,x₃} is the union of the three geodesics [x₁x₂], [x₂x₃] and [x₃x₁] in X. The space X is δ-hyperbolic (in the Gromov sense) if, for every geodesic triangle T in X, every side of T is contained in a δ-neighborhood of the union of the other two sides. We denote by δ(X) the sharpest hyperbolicity constant of X, i.e. . In this paper, we obtain several tight bounds for the hyperbolicity constant of a graph and precise values of this constant for some important families of graphs. In particular, we investigate the relationship between the hyperbolicity constant of a graph and its number of edges, diameter and cycles. As a consequence of our results, we show that if G is any graph with m edges with lengths , then , and if and only if G is isomorphic to C_m. Moreover, we prove the inequality for every graph, and we use this inequality in order to compute the precise value δ(G) for some common graphs.     

Bipancyclic properties of Cayley graphs generated by transpositions

Cycle is one of the most fundamental graph classes. For a given graph, it is interesting to find cycles of various lengths as subgraphs in the graph. The Cayley graph on the symmetric group has an important role for the study of Cayley graphs as interconnection networks. In this paper, we show that the Cayley graph generated by a transposition set is vertex-bipancyclic if and only if it is not the star graph. We also provide a necessary and sufficient condition for to be edge-bipancyclic.