Graph Bundles with productive fibers when their voltages lie in the product of Cayley graphs

Kim, Hye Kyung 대구효성가톨릭대학교 자연과학연구소 1998 基礎科學硏究論集 Vol.12 No.-

Kwak and Lee [4] computed the characteristic polynomial of a graph bundle(also, a graph covering) when its voltage lie in an abelian subgroup of the automorphism group of the fiber; in particular, the automorphism group of the fiber is abelian. [3], I also computed thecharacteristic polynomial of a graph bundles with productive fibers when their voltages lie in the product of abelian groups. In this paper, 1 obtain some formulars for computing the characteristic polynomial of a graph bundles with productive fibers when their voltages lie in the product of Cayley graphs. This result is more explicit than those at [3]. Key words : graph bundle, cayley graph.

LAPLACIAN SPECTRA OF GRAPH BUNDLES

Kim, Ju-Young Korean Mathematical Society 1996 대한수학회논문집 Vol.11 No.4

The spectrum of the Laplacian matrix of a graph gives an information of the structure of the graph. For example, the product of non-zero eigenvalues of the characteristic polynomial of the Laplacian matrix of a graph with n vertices is n times of the number of spanning trees of that graph. The characteristic polynomial of the Laplacian matrix of a graph tells us the number of spanning trees and the connectivity of given graph. in this paper, we compute the characteristic polynomial of the Laplacian matrix of a graph bundle when its voltage lie in an abelian subgroup of the full automorphism group of the fibre; in particular, the automorphism group of the fibre is abelian. Also we study a relation between the characteristic polynomial of the Laplacian matrix of a graph G and that of the Laplacian matrix of a graph bundle over G. Some applications are also discussed.

Bartholdi zeta and <i>L</i>-functions of weighted digraphs, their coverings and products

Choe, Young-Bin,Kwak, Jin Ho,Park, Yong Sung,Sato, Iwao Elsevier 2007 Advances in mathematics Vol.213 No.2

<P><B>Abstract</B></P><P>Since a zeta function of a regular graph was introduced by Ihara [Y. Ihara, On discrete subgroups of the two by two projective linear group over <I>p</I>-adic fields, J. Math. Soc. Japan 19 (1966) 219–235], many kinds of zeta functions and <I>L</I>-functions of a graph or a digraph have been defined and investigated. Most of the works concerning zeta and <I>L</I>-functions of a graph contain the following: (1) defining a zeta function, (2) defining an <I>L</I>-function associated with a (regular) graph covering, (3) providing their determinant expressions, and (4) computing the zeta function of a graph covering and obtaining its decomposition formula as a product of <I>L</I>-functions. As a continuation of those works, we introduce a zeta function of a weighted digraph and an <I>L</I>-function associated with a weighted digraph bundle. A graph bundle is a notion containing a cartesian product of graphs and a (regular or irregular) graph covering. Also we provide determinant expressions of the zeta function and the <I>L</I>-function. Moreover, we compute the zeta function of a weighted digraph bundle and obtain its decomposition formula as a product of the <I>L</I>-functions.</P>

ON DOMINATION NUMBERS OF GRAPH BUNDLES

Zmazek, Blaz,Zerovnik, Janez 한국전산응용수학회 2006 Journal of applied mathematics & informatics Vol.22 No.1

Let ${\gamma}$(G) be the domination number of a graph G. It is shown that for any $k {\ge} 0$ there exists a Cartesian graph bundle $B{\Box}_{\varphi}F$ such that ${\gamma}(B{\Box}_{\varphi}F) ={\gamma}(B){\gamma}(F)-2k$. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing's conjecture on strong graph bundles is shown not to be true by proving the inequality ${\gamma}(B{\bigotimes}_{\varphi}F){\le}{\gamma}(B){\gamma}(F)$ for strong graph bundles. Examples of graphs Band F with ${\gamma}(B{\bigotimes}_{\varphi}F) < {\gamma}(B){\gamma}(F)$ are given.

Perfect domination sets in Cayley graphs

Kwon, Y.S.,Lee, J. North Holland ; Elsevier Science Ltd 2014 Discrete Applied Mathematics Vol.162 No.-

In this paper, we get some results related to perfect domination sets of Cayley graphs. We show that if a Cayley graph C(A,X) has a perfect dominating set S which is a normal subgroup of A and whose induced subgraph is F, then there exists an F-bundle projection p:C(A,X)→K<SUB>m</SUB> for some positive integer m. As an application, we show that for any positive integer n, the following are equivalent: (a) the hypercube Q<SUB>n</SUB> has a perfect total domination set, (b) n=2<SUP>m</SUP> for a positive integer m, (c) Q<SUB>n</SUB> is a 2<SUP>n-log'2n-1</SUP>K<SUB>2</SUB>-bundle over the complete graph K<SUB>n</SUB> and (d) Q<SUB>n</SUB> is a covering of the complete bipartite graph K<SUB>n,n</SUB>.

ZETA FUNCTIONS OF GRAPH BUNDLES

Feng, Rongquan,Kwak, Jin-Ho Korean Mathematical Society 2006 대한수학회지 Vol.43 No.6

As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle.

CHARACTERISTIC POLYNOMIALS OF SOME WEIGHTED GRAPH BUNDLES AND ITS APPLICATION TO LINKS

Sohn, Moo-Young,Lee, Ja-Eun 國立 昌原大學校 基礎科學硏究所 1994 基礎科學硏究所論文集 Vol.6 No.-

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K_(2) (??_(2)) bundles over a weighted graph Γ_(ω) can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are Γ As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

CHARACTERISTIC POLYNOMIALS OF SOME WEIGHTED GRAPH BUNDLES AND ITS APPLICATION TO LNKS

SOHN, MOO YOUNG,LEE, JAEUN 경북대학교 위상수학 기하학연구센터 1995 硏究論文集 Vol.1 No.-

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K_(2) (??_(2))- bundles over a weighted graph Γ_(ω) can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are Γ As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

CHARACTERISTIC POLYNOMIALS OF SOME WEIGHTED GRAPH BUNDLES AND ITS APPLICATION TO LINKS

LEE,JAEUN,SOHN,MOO YOUNG 國立昌原大學校 基礎科學硏究所 1994 基礎科學硏究所論文集 Vol.6 No.-

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K₂(K₂)-bundles over a weighted graph Γω can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are Γ As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

Cayley그래프들를 파이버로 가지는 그래프 다발들의 특성방정식들

김혜경 대구효성가톨릭대학교 1998 연구논문집 Vol.57 No.2

논문[4]에서 전압함수들이 파이버의 동형군의 가환 부분군 안에 있을때의 그래프 다발들의 특성방정식들의 공식을 구하였으며 특히 파이버의 동형군이 가환일때 구했다. 이 논문에서는 임의의 가환군 Γ에 대하여 파이버가 임의의 Cayley 그래프들 인 그래프 다발들의 특성방정식들을 논문[4]에서 보다 더욱 명확한 공식을 계산했다. Kwak and Lee [4] computed the characteristic polynomial of a graph bundle (also, a graph covering) when its voltage lie in an abelian subgroup of the automorphism group of the fiber; in particular, the automorphism group of the fiber is abelian. In this paper, for any abelian group Γand any Cayley graph Cay(Γ,T), we obtain some formulas for computing the characteristic polynomials of Cay(Γ,T)-bundles which are more explicit than those at Kwak and Lee's.