Jaeun Lee

Regular graph coverings whose covering transformation groups have the isomorphism extension property

Abstract Enumerative results are presently a major center of interest in topological graph theory, as in the work of Gross and Furst [1], Hofmeister [5,6], Kwak and Lee [9–13] and Mull et al. [15], etc. Kwak and Lee [9] enumerated the isomorphism classes of graph bundles and those of n -fold graph coverings with respect to a group of automorphisms of the base graph which fix a spanning tree. Hofmeister [6] enumerated independently the isomorphism classes of n -fold graph coverings with respect to the trivial automorphism group of the base graph. But the enumeration of isomorphism classes of regular graph coverings has not been answered completely. As its partial answers, Hofmeister enumerated the isomorphism classes of Z 2 -coverings (double coverings) with respect to any group of automorphisms of the base graph, and Sato [14] did the same work for Z p -coverings (regular prime-fold coverings). With respect to the trivial automorphism group of the base graph, Hong and Kwak [8] did the same work for Z 2 ⊕ Z 2 or Z 4 -coverings, and Kwak and Lee [10] did it for Z p , Z p ⊕ Z 1 ( p ≠ q primes) or Z p 2 -coverings. As an expansion of this effort, we obtain in this paper several new algebraic characterizations for isomorphic regular coverings and derive an enumerating formula for the isomorphism classes of A-coverings of a graph G with respect to any group of automorphisms of G which fix a spanning tree, when the covering transformation group A has the isomorphism extension property. By definition, it means that every isomorphism between any two isomorphic subgroups B 1 and B 2 of A can be extended to an automorphism of A. Also, we obtain complete numerical enumeration of the isomorphism classes of Z n -coverings for all n , D n -coverings for odd n (D n is the dihedral group of order 2 n ) or Z p ⊕ Z p -coverings of a graph G for prime p with respect to the trivial automorphism group of G . In addition, we applied our results to a bouquet of circles.

Enumeration of graph embeddings

Abstract For a finite connected simple graph G , let Γ be a group of graph automorphisms of G . Two 2-cell embeddings ι: G → S and j : G → S of a graph G into a closed surface S (orientable or nonorientable) are congruent with respect to Γ if there are a surface homeomorphism h : S → S and a graph automorphism γϵΓ such that h oι= j oγ. In this paper, we give an algebraic characterization of congruent 2-cell embeddings, from which we enumerate the congruence classes of 2-cell embeddings of a graph G into closed surfaces with respect to a group of automorphisms of G , not just the full automorphism group. Some applications to complete graphs are also discussed. As an orientable case, the oriented congruence of a graph G into orientable surfaces with respect to the full automorphism group of G was enumerated by Mull et al. (1988).

Caracteristics polynomials of some grap bundlesII

The characteristic polynomial of a graph G is that of its adjacency matrix, and its eigenvalues are those of its adjacency matrix. Recently, Y. Chae, J. H. Kwak and J. Lee showed a relation between the characteristic polynomial of a graph G and those of graph bundles over G. In particular, the characteristic polynomial of G is a divisor of those of its covering graphs. They also gave the complete computation of the characteristic polynomials of K 2 ( )-bundles over a graph. In this paper, we compute the characteristic polynomial of a graph bundle when its voltages lie in an abelian subgroup of the full automorphism group of the fibre; in particular, the automorphism group of the fibre is abelian. Some applications to path- or cycle-bundles are also discussed.

Some invariants of Pretzel links

We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of n-pretzel links using these polynomials and find the basket number of pretzel links by showing that the genus and the canonical genus of a pretzel link are the same.

Enumeration of Regular Graph Coverings Having Finite Abelian Covering Transformation Groups

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors. An enumeration of the isomorphism classes of n-fold coverings of a graph G was done by Kwak and Lee [Canad. J. Math., XLII (1990), pp. 747--761] and independently by Hofmeister [Discrete Math., 98 (1991), pp. 437--444]. An enumeration of the isomorphism classes of connected n-fold coverings of a graph G was recently done by Kwak and Lee [J. Graph Theory, 23 (1996), pp. 105--109]. But the enumeration of the isomorphism classes of regular coverings of a graph G has been done for only a few cases. In fact, the isomorphism classes of

Distributions of regular branched prime-fold coverings of surfaces

{\cal A}

Counting some finite-fold coverings of a graph

-coverings of G were enumerated when

Bipartite covering graphs

{\cal A}

Characteristic polynomials of graph coverings

is the cyclic group

Isomorphism classes of cycle permutation graphs

\BZ_n