Classification of Regular Embeddings of Complete Multipartite Graphs

A 2‐cell embedding of a graph Γ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags. In this article, we classify the regular embeddings of the complete multipartite graph $K_{n,\dots ,n}$. We show that if the number of partite sets is greater than 3, there exists no such embedding; and if the number of partite sets is 3, for any n, there exist one orientable regular embedding and one nonorientable regular embedding of $K_{n,n,n}$ up to isomorphism.     

Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs

In this paper, we classify the reflexible regular orientable embeddings and the self-Petrie dual regular orientable embeddings of complete bipartite graphs. The classification shows that for any natural number n, say (p₁,p₂,…,p_k are distinct odd primes and a_i>0 for each i?1), there are t distinct reflexible regular embeddings of the complete bipartite graph K_n,n up to isomorphism, where t=1 if a=0, t=2^k if a=1, t=2^k+1 if a=2, and t=3·2^k+1 if a?3. And, there are s distinct self-Petrie dual regular embeddings of K_n,n up to isomorphism, where s=1 if a=0, s=2^k if a=1, s=2^k+1 if a=2, and s=2^k+2 if a?3.     

Self-dual embeddings of complete multipartite graphs

In this paper we examine self-dual embeddings of complete multipartite graphs, focusing primarily on K_m(n) having m parts each of size n. If m = 2, then n must be even. If the embedding is on an orientable surface, then an Euler characteristic argument shows that no such embedding exists when n is odd and m ? 2, 3 (mod 4); there is no such restriction for embeddings on nonorientable surfaces. We show that these embeddings exist with a few small exceptions. As a corollary, every group has a Cayley graph with a self-dual embedding. Our main technique is an addition construction that combines self-dual embeddings of two subgraphs into a self-dual embedding of their union. We also apply this technique to nonregular multipartite graphs and to cubes.     

Classification of regular embeddings of <Emphasis Type="Italic">n</Emphasis>-dimensional cubes

An orientably-regular map is a 2-cell embedding of a connected graph or multigraph into an orientable surface, such that the group of all orientation-preserving automorphisms of the embedding has a single orbit on the set of all arcs (incident vertex-edge pairs). Such embeddings of the n-dimensional cubes Q _n were classified for all odd n by Du, Kwak and Nedela in 2005, and in 2007, Jing Xu proved that for n=2m where m is odd, they are precisely the embeddings constructed by Kwon in 2004. Here, we give a classification of orientably-regular embeddings of Q _n for all n. In particular, we show that for all even n (=2m), these embeddings are in one-to-one correspondence with elements σ of order 1 or 2 in the symmetric group S _n such that σ fixes n, preserves the set of all pairs B _i ={i,i+m} for 1≤i≤m, and induces the same permutation on this set as the permutation B _i ↦ B _f(i) for some additive bijection f:ℤ _m →ℤ _m . We also give formulae for the numbers of embeddings that are reflexible and chiral, respectively, showing that the ratio of reflexible to chiral embeddings tends to zero for large even n.     

New regular embeddings of n‐cubes Qn

We investigate highly symmetrical embeddings of the n‐dimensional cube Q_n into orientable compact surfaces which we call regular embeddings of Q_n. We derive some general results and construct some new families of regular embeddings of Q_n. In particular, for n = 1, 2,3(mod 4), we classify the regular embeddings of Q_n which can be expressed as balanced Cayley maps. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 297–312, 2004     

The ordering of trees and connected graphs by algebraic connectivity

In this paper, we first determine that the first four trees of order n?9 with the smallest algebraic connectivity are P_n,Q_n,W_n and Z_n with α(P_n)<α(Q_n)<α(W_n)<α(Z_n)<α(T), where T is any tree of order n other than P_n, Q_n, W_n, and Z_n. Then we consider the effect on the Laplacian eigenvalues of connected graphs by suitably adding edges. By using these results and methods, we finally determine that the first six connected graphs of order n?9 with the smallest algebraic connectivity are and , with , where G is any connected graph of order n other than P_n, Q_n, , W_n, and .     

Spanning multi-paths in hypercubes

A set A of vertices of a hypercube is called balanced if . We prove that for every natural number n there exists a natural number π₁(n) such that for every hypercube Q with dim(Q)?π₁(n) there exists a family of pairwise vertex-disjoint paths P_i between A_i and B_i for i=1,2,…,n with if and only if {A_i,B_i∣i=1,2,…,n} is a balanced set.     

A new approach to constructing exponentially many nonisomorphic nonorientable triangular embeddings of complete graphs

We prove a theorem that for an integer s?0, if 12s+7 is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of K_n, where n=(12s+7)(6s+7), is at least . By some number-theoretic arguments there are an infinite number of integers s satisfying the hypothesis of the theorem. The theorem is the first known example of constructing at least 2^αn?+o(n?), ?>1, nonisomorphic nonorientable triangular embeddings of K_n for n=6t+1, . To prove the theorem, we use a new approach to constructing nonisomorphic triangular embeddings of complete graphs. The approach combines a cut-and-paste technique and the index one current graph technique. A new connection between Steiner triple systems and constructing triangular embeddings of complete graphs is given.     

A spectral condition for odd cycles in graphs

Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ(G) of its adjacency matrix satisfies . Then G contains a cycle of length t for every t?n/320This condition is sharp: the complete bipartite graph T₂(n) with parts of size ⌊n/2⌋ and ⌈n/2⌉ contains no odd cycles and its largest eigenvalue is equal to .This condition is stable: if μ(G) is close to and G fails to contain a cycle of length t for some t?n/321, then G resembles T₂(n).     

Maximum genus embeddings of Steiner triple systems

We prove that for n>3 every STS(n) has both an orientable and a nonorientable embedding in which the triples of the STS(n) appear as triangular faces and there is just one additional large face. We also obtain detailed results about the possible automorphisms of such embeddings.     

On the number of representations of certain integers as sums of 11 or 13 squares

Let r_k(n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p,

     

Isometric embeddings of real projective spaces into Euclidean spaces

This paper studies isometric embeddings of RPn via non-degenerate symmetric bilinear maps. The main result shows the infimum dimension of target Euclidean spaces among these constructions for RPn is . Next, we construct Veronese maps by induction, which realize the infimum. Finally, we give a simple proof of Rigidity Theorem of Veronese maps.     

On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with no CM and a fixed modular parametrization and let be Heegner points attached to the rings of integers of distinct quadratic imaginary fields k₁,…,kr. We prove that if the odd parts of the class numbers of k₁,…,kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in .     

Semisymmetric graphs

Let be a regular covering projection of connected graphs with the group of covering transformations isomorphic to N. If N is an elementary abelian p-group, then the projection ℘_N is called p-elementary abelian. The projection ℘_N is vertex-transitive (edge-transitive) if some vertex-transitive (edge-transitive) subgroup of the automorphism group of X lifts along ℘_N, and semisymmetric if it is edge- but not vertex-transitive. The projection ℘_N is minimal semisymmetric if it cannot be written as a composition ℘_N=℘℘_M of two (nontrivial) regular covering projections, where ℘_M is semisymmetric.Malni? et al. [Semisymmetric elementary abelian covers of the Möbius-Kantor graph, Discrete Math. 307 (2007) 2156-2175] determined all pairwise nonisomorphic minimal semisymmetric elementary abelian regular covering projections of the Möbius-Kantor graph, the Generalized Petersen graph GP(8,3), by explicitly giving the corresponding voltage rules generating the covering projections. It was remarked at the end of the above paper that the covering graphs arising from these covering projections need not themselves be semisymmetric (a graph with regular valency is said to be semisymmetric if its automorphism group is edge- but not vertex-transitive). In this paper it is shown that all these covering graphs are indeed semisymmetric.     

Minimum genus embeddings of the complete graph

In this paper,the problem of construction of exponentially many minimum genus embeddings of complete graphs in surfaces are studied.There are three approaches to solve this problem.The first approach is to construct exponentially many graphs by the theory of graceful labeling of paths;the second approach is to find a current assignment of the current graph by the theory of current graph;the third approach is to find exponentially many embedding(or rotation) schemes of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph.According to this three approaches,we can construct exponentially many minimum genus embeddings of complete graph K_(12s+8) in orientable surfaces,which show that there are at least 10/3×(200/9)~s distinct minimum genus embeddings for K_(12s+8) in orientable surfaces.We have also proved that K_(12s+8) has at least 10/3×(200/9)~s distinct minimum genus embeddings in non-orientable surfaces.     

On Estrada index of trees

Let G be a graph on n vertices, and let λ₁,λ₂,…,λ_n be its eigenvalues. The Estrada index is defined as . We determine the unique tree with maximum Estrada index among the trees on n vertices with given matching number, and the unique tree with maximum Estrada index among the trees on n vertices with fixed diameter. For , we also determine the tree with maximum Estrada index among the trees on n vertices with maximum degree Δ. It gives a partial solution to the conjecture proposed by Ili? and Stevanovi? in Ref. [14].     

Regular orientable embeddings of complete bipartite graphs

In this paper, it will be shown that the isomorphism classes of regular orientable embeddings of the complete bipartite graph K_n,n are in one‐to‐one correspondence with the permutations on n elements satisfying a given criterion, and the isomorphism classes of them are completely classified when n is a product of any two (not necessarily distinct) prime numbers. For other n, a lower bound of the number of those isomorphism classes of K_n,n is obtained. As a result, many new regular orientable embeddings of the complete bipartite graph are constructed giving an answer of Nedela‐?koviera's question raised in 12 . © 2005 Wiley Periodicals, Inc. J Graph Theory     

A family of invariants of rooted forests

Let A be a commutative k-algebra over a field of k and Ξ a linear operator defined on A. We define a family of A-valued invariants Ψ for finite rooted forests by a recurrent algorithm using the operator Ξ and show that the invariant Ψ distinguishes rooted forests if (and only if) it distinguishes rooted trees T, and if (and only if) it is finer than the quantity α(T)=|Aut(T)| of rooted trees T. We also consider the generating function with , where is the set of rooted trees with n vertices. We show that the generating function U(q) satisfies the equation . Consequently, we get a recurrent formula for U_n (n?1), namely, U₁=Ξ(1) and U_n=ΞS_n−1(U₁,U₂,…,U_n−1) for any n?2, where are the elementary Schur polynomials. We also show that the (strict) order polynomials and two well-known quasi-symmetric function invariants of rooted forests are in the family of invariants Ψ and derive some consequences about these well-known invariants from our general results on Ψ. Finally, we generalize the invariant Ψ to labeled planar forests and discuss its certain relations with the Hopf algebra in Foissy (Bull. Sci. Math. 126 (2002) 193) spanned by labeled planar forests.