期刊：
Journal of Algebraic Combinatorics,2021年53(2):435-454 ISSN：0925-9899
通讯作者：
Xu, Bangteng
作者机构：
[Chen, Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Xu, Bangteng] Eastern Kentucky Univ, Dept Math & Stat, Richmond, KY 40475 USA.
通讯机构：
[Xu, Bangteng] E;Eastern Kentucky Univ, Dept Math & Stat, Richmond, KY 40475 USA.
关键词：
Association schemes;Thin residues;Automorphism groups;Schurian schemes;\(O^{\vartheta }\)(S)-feasible association schemes
期刊：
JOURNAL OF COMBINATORIAL THEORY SERIES B,2019年139:128-162 ISSN：0095-8956
通讯作者：
Jing, Guangming
作者机构：
[Chen, Guantao; Jing, Guangming] Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.;[Chen, Guantao] Cent China Normal Univ, Sch Math & Stat, Wuhan, Hubei, Peoples R China.
通讯机构：
[Jing, Guangming] G;Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.
关键词：
Chromatic index;Graph density;Critical chromatic graph;Tashkinov tree
摘要：
Let G be a graph with possible multiple edges but no loops. The density of G, denoted by rho(G) , is defined as max(H subset of G, vertical bar V(H)vertical bar >= 2) inverted right perpendicular vertical bar E(H)vertical bar/left perpendicular vertical bar V(H)vertical bar/2 right perpendicular inverted left perpendicular. Goldberg (1973) and Seymour (1974) independently conjectured that if the chromatic index chi'(G) satisfies chi'(G) >= Delta(G) + 2 then chi'(G) = rho(G), which is commonly regarded as Goldberg's conjecture. An equivalent conjecture, usually credited to Jakobsen, states that for any odd integer m >= 3, if chi'(G) >= m Delta(G)/m-1 + m-3/m-1 then chi'(G) = rho(G). The Tashkinov tree technique, a common generalization of Vizing fans and Kierstead paths for multigraphs, has emerged as the main tool to attack these two conjectures. On the other hand, Asplund and McDonald recently showed that there is a limitation to this method. In this paper, we will go beyond Tashkinov trees and provide a much larger extended structure, using which we see hope to tackle the conjecture. Applying this new technique, we show that the Goldberg's conjecture holds for graphs with Delta(G) <= 39, or vertical bar V(G)vertical bar <= 39 and the Jakobsen Conjecture holds for m <= 39, where the previously known best bound is 23. We also improve a number of other related results. (C) 2019 Elsevier Inc. All rights reserved.
摘要：
Given a graph $G$, denote by $\Delta$, $\bar{d}$ and $\chi^\prime$ the maximum degree, the average degree and the chromatic index of $G$, respectively. A simple graph $G$ is called {\it edge-$\Delta$-critical} if $\chi^\prime(G)=\Delta+1$ and $\chi^\prime(H)\le\Delta$ for every proper subgraph $H$ of $G$. Vizing in 1968 conjectured that if $G$ is edge-$\Delta$-critical, then $\bar{d}\geq \Delta-1+ \frac{3}{n}$. We show that $$ \begin{displaystyle} \avd \ge \begin{cases} 0.69241\D-0.15658 \quad\,\: \mbox{ if } \Delta\geq 66, 0.69392\D-0.20642\quad\;\,\mbox{ if } \Delta=65, \mbox{ and } 0.68706\D+0.19815\quad\! \quad\mbox{if } 56\leq \Delta\leq64. \end{cases} \end{displaystyle} $$ This result improves the best known bound $\frac{2}{3}(\Delta +2)$ obtained by Woodall in 2007 for $\Delta \geq 56$. Additionally, Woodall constructed an infinite family of graphs showing his result cannot be improved by well-known Vizing's Adjacency Lemma and other known edge-coloring techniques. To over come the barrier, we follow the recently developed recoloring technique of Tashkinov trees to expand Vizing fans technique to a larger class of trees.
作者机构：
[Chen, Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Xu, Bangteng] Eastern Kentucky Univ, Dept Math & Stat, Richmond, KY 40475 USA.
通讯机构：
[Xu, Bangteng] E;Eastern Kentucky Univ, Dept Math & Stat, Richmond, KY 40475 USA.
关键词：
Character tables;Closed subsets;Generalized wreath products;Table algebras;Association schemes
摘要：
Structures of table algebras whose character tables have a zero submatrix were studied in Blau and Xu (2014) and Chen and Xu (2018). In this paper we continue the research in this direction, and investigate the conditions under which the character table with a zero submatrix yields a generalized wreath product of table algebras. Applications to association schemes are also discussed. (C) 2018 Elsevier B.V. All rights reserved.
摘要：
Graph edge coloring has a rich theory, many applications and beautiful conjectures, and it is studied not only by mathematicians, but also by computer scientists. In this survey, written for the non-expert, we shall describe some main results and techniques and state some of the many popular conjectures in the theory. Besides known results a new basic result about brooms is obtained.
期刊：
Communications in Algebra,2018年46(8):3510-3519 ISSN：0092-7872
通讯作者：
Xu, Bangteng
作者机构：
[Chen, Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan, Hubei, Peoples R China.;[Xu, Bangteng] Eastern Kentucky Univ, Dept Math & Stat, Richmond, KY 40475 USA.
通讯机构：
[Xu, Bangteng] E;Eastern Kentucky Univ, Dept Math & Stat, Richmond, KY 40475 USA.
关键词：
Character tables;closed subsets;dual C-algebras;finite groups;generalized wreath products;table algebras
摘要：
ABSTRACT Structures of table algebras determined by their character tables are studied in [6 Blau, H., Xu, B. (2014). Irreducible characters of wreath products in reality-based algebras and applications to association schemes. J. Algebra 412 (2014), 155–172.[Crossref] , [Google Scholar]]. In this paper we continue the research in this direction. In particular, we investigate the conditions under which the character table with a zero submatrix yields a generalized wreath product for a commutative table algebra. Applications to finite groups are also discussed, and some new and known results are obtained as direct consequences.
期刊：
Journal of Algebra,2017年488:201-229 ISSN：0021-8693
通讯作者：
Ponomarenko, Ilia
作者机构：
[Chen, Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan, Hubei, Peoples R China.;[Ponomarenko, Ilia] Steklov Inst Math, St Petersburg, Russia.
通讯机构：
[Ponomarenko, Ilia] S;Steklov Inst Math, St Petersburg, Russia.
摘要：
Let X be a coherent configuration associated with a transitive group G. In terms of the intersection numbers of X, a necessary condition for the point stabilizer of G to be a TI-subgroup, is established. Furthermore, under this condition, X is determined up to isomorphism by the intersection numbers. It is also proved that asymptotically, this condition is also sufficient. More precisely, an arbitrary homogeneous coherent configuration satisfying this condition is associated with a transitive group, the point stabilizer of which is a TI-subgroup. As a byproduct of the developed theory, recent results on pseudocyclic and quasi-thin association schemes are generalized and improved. In particular, it is shown that any scheme of prime degree p and valency k is associated with a transitive group, whenever p > 1 + 6k(k - 1)(2). (C) 2017 Elsevier Inc. All rights reserved.
期刊：
SIAM JOURNAL ON DISCRETE MATHEMATICS,2017年31(4):2429-2439 ISSN：0895-4801
通讯作者：
Chen, Guantao
作者机构：
[Chen, Guantao] Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.;[Chen, Guantao] Cent China Normal Univ, Fac Math & Stat, Wuhan, Hubei, Peoples R China.;[Enomoto, Hikoe] Waseda Univ, Grad Sch Econ, Tokyo 1698050, Japan.;[Ozeki, Kenta] Yokohama Natl Univ, Fac Environm & Informat Sci, Yokohama, Kanagawa 2408501, Japan.;[Tsuchiya, Shoichi] Senshu Univ, Sch Network & Informat, Kawasaki, Kanagawa 2148580, Japan.
通讯机构：
[Chen, Guantao] G;[Chen, Guantao] C;Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.;Cent China Normal Univ, Fac Math & Stat, Wuhan, Hubei, Peoples R China.
期刊：
Graphs and Combinatorics,2017年33(5):1321-1345 ISSN：0911-0119
通讯作者：
Tsuchiya, Shoichi
作者机构：
[Chen, Guantao] Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.;[Chen, Guantao] Cent China Normal Univ, Fac Math & Stat, Wuhan, Hubei, Peoples R China.;[Han, Jie] Univ Sao Paulo, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo, Brazil.;[Suil, O.] SUNY, Dept Appl Math & Stat, Incheon 21985, South Korea.;[Shan, Songling] Vanderbilt Univ, Dept Math, Nashville, TN 37240 USA.
摘要：
A Halin graph is constructed from a plane embedding of a tree with no vertices of degree 2 by adding a cycle through its leaves in the natural order determined by the embedding. Halin graphs satisfy interesting properties. However, to our knowledge, there are no results giving a positive answer for "spanning Halin subgraph problem" (i.e., which graph has a Halin graph as a spanning subgraph) except for a conjecture by Lovasz and Plummer which states that every 4-connected plane triangulation contains a spanning Halin subgraph. In this paper, we investigate the characterization of forbidden pairs guaranteeing the existence of a spanning Halin subgraph. In particular, we show that the set of such pairs is a very small class. Also, we show that belongs to the set, but neither nor belongs to the set.
作者机构：
[Chen, Guantao] Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.;[Chen, Guantao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Chen, Xiaodong] Liaoning Univ Technol, Coll Sci, Jinzhou 121001, Peoples R China.;[Zhao, Yue] Univ Cent Florida, Dept Math, Orlando, FL 32816 USA.
期刊：
Algebra and Logic,2016年55(1):38-49 ISSN：0002-5232
通讯作者：
Muzychuk, M. E.
作者机构：
[Muzychuk, M. E.] Netanya Acad Coll, Netanya, Israel.;[Ponomarenko, I. N.] VA Steklov Math Inst, St Petersburg Branch, St Petersburg 191011, Russia.;[Chen, G.] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.
通讯机构：
[Muzychuk, M. E.] N;Netanya Acad Coll, Netanya, Israel.
关键词：
S-ring;conjugacy class;B-group
摘要：
Two basic results on S-rings over an Abelian group are the Schur theorem on multipliers and the Wielandt theorem on primitive S-rings over groups with a cyclic Sylow subgroup. Neither of these is directly generalized to the non-Abelian case. Nevertheless, we prove that the two theorems are true for central S-rings over any group, i.e., for S-rings that are contained in the center of the group ring of that group (such S-rings arise naturally in the supercharacter theory). Extending the concept of a B-group introduced by Wielandt, we show that every Camina group is a generalized B-group, whereas simple groups, with few exceptions, cannot be of this type.
期刊：
Archiv der Mathematik,2015年105(4):313-322 ISSN：0003-889X
通讯作者：
Blau, Harvey I.
作者机构：
[Blau, Harvey I.] No Illinois Univ, Dept Math Sci, De Kalb, IL 60115 USA.;[Chen, Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Blau, Harvey I.] N;No Illinois Univ, Dept Math Sci, De Kalb, IL 60115 USA.
摘要：
Table algebras all of whose nonidentity basis elements are involutions (in the sense of Zieschang), which serve as a counterpoint to the generic Hecke algebras parametrized by Coxeter groups, are classified. If two-generated, they are the family H (n) (for all n a parts per thousand yen 3), which for suitable n arise from schemes defined by affine planes of order n - 1. Otherwise, the basis involutions correspond to the points of a finite projective space whose incidence geometry determines the algebra multiplication. This generalizes to table algebras a previous result of van Dam for association schemes. An algebraic characterization is also given.
期刊：
Algebras and Representation Theory,2014年17(5):1527-1552 ISSN：1386-923X
通讯作者：
Blau, Harvey I.
作者机构：
[Blau, Harvey I.] No Illinois Univ, Dept Math Sci, De Kalb, IL 60115 USA.;[Chen, Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Blau, Harvey I.] N;No Illinois Univ, Dept Math Sci, De Kalb, IL 60115 USA.
关键词：
Table algebras;Union of closed subsets;Association schemes;Finite groups;Irreducible characters
期刊：
Journal of Graph Theory,2014年77(3):237-250 ISSN：0364-9024
通讯作者：
Chen, Guantao
作者机构：
[Chen, Guantao] Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.;[Ferrara, Michael; Jacobson, Michael] Univ Colorado Denver, Dept Math & Stat Sci, Denver, CO USA.;[Hu, Zhiquan] Cent China Normal Univ, Fac Math & Stat, Wuhan, Peoples R China.;[Liu, Huiqing] Hubei Univ, Sch Math & Comp Sci, Wuhan, Peoples R China.
通讯机构：
[Chen, Guantao] G;Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA.
关键词：
extremal spanning tree;broom
摘要：
A broom is a tree obtained by subdividing one edge of the star an arbitrary number of times. In (E. Flandrin, T. Kaiser, R. Kužel, H. Li and Z. Ryjáček, Neighborhood Unions and Extremal Spanning Trees, Discrete Math 308 (2008), 2343–2350) Flandrin et al. posed the problem of determining degree conditions that ensure a connected graph G contains a spanning tree that is a broom. In this article, we give one solution to this problem by demonstrating that if G is a connected graph of order with , then G contains a spanning broom. This result is best possible.
期刊：
Journal of Algebra,2013年373:340-355 ISSN：0021-8693
通讯作者：
Blau, Harvey I.
作者机构：
[Blau, Harvey I.] No Illinois Univ, Dept Math Sci, De Kalb, IL 60115 USA.;[Chen, Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Blau, Harvey I.] N;No Illinois Univ, Dept Math Sci, De Kalb, IL 60115 USA.
关键词：
C-algebras;Table algebras;Algebras with involution;Finite groups;Irreducible characters;Conjugacy classes;Matrices;Dual C-algebra;Association schemes
摘要：
A lower bound on the number of elements outside a closed subset of a C-basis of a commutative C-algebra with dual a table algebra is derived, as is an equivalent condition for when the lower bound is met. As corollaries, lower bounds are obtained on the number of primitive idempotent matrices of rank greater than 1 in the adjacency algebra of a commutative, imprimitive association scheme; and, for a given normal subgroup N of a finite group C. on the number of irreducible characters of G whose kernels do not contain N, and on the number of conjugacy classes of G not contained in N. Also found are equivalent conditions for when these lower bounds are met. (C) 2012 Elsevier Inc. All rights reserved.
期刊：
Communications in Algebra,2013年41(6):2387-2404 ISSN：0092-7872
通讯作者：
Chen, Gang
作者机构：
[Blau, Harvey I.] No Illinois Univ, Dept Math Sci, De Kalb, IL USA.;[Chen, Gang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Chen, Gang] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要：
It is proved that nonvanishing elements of a commutative nilpotent table algebra must be linear, which generalizes the known result on the nonvanishing elements of a finite nilpotent group. Other results on nonvanishing elements of finite groups are generalized to the context of a commutative table algebra whose dual is also a commutative table algebra.
