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On the Numbers of Cut-Vertices and End-Blocks in 4-Regular Graphs

Dingguo WangErfang Shan — 2014

Discussiones Mathematicae Graph Theory

A cut-vertex in a graph G is a vertex whose removal increases the number of connected components of G. An end-block of G is a block with a single cut-vertex. In this paper we establish upper bounds on the numbers of end-blocks and cut-vertices in a 4-regular graph G and claw-free 4-regular graphs. We characterize the extremal graphs achieving these bounds.

The bicrossed products of H 4 and H 8

Daowei LuYan NingDingguo Wang — 2020

Czechoslovak Mathematical Journal

Let H 4 and H 8 be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through H 8 and H 4 (equivalently, any bicrossed product between the Hopf algebras H 8 and H 4 ) must be isomorphic to one of the following four Hopf algebras: H 8 H 4 , H 32 , 1 , H 32 , 2 , H 32 , 3 . The set of all matched pairs ( H 8 , H 4 , , ) is explicitly described, and then the associated bicrossed product is given by generators and relations.

Bicrossed products of generalized Taft algebra and group algebras

Dingguo WangXiangdong ChengDaowei Lu — 2022

Czechoslovak Mathematical Journal

Let G be a group generated by a set of finite order elements. We prove that any bicrossed product H m , d ( q ) k [ G ] between the generalized Taft algebra H m , d ( q ) and group algebra k [ G ] is actually the smash product H m , d ( q ) k [ G ] . Then we show that the classification of these smash products could be reduced to the description of the group automorphisms of G . As an application, the classification of H m , d ( q ) k [ C n 1 × C n 2 ] is completely presented by generators and relations, where C n denotes the n -cyclic group.

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