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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.

In this paper, we study the $H^{\sigma -R}$ type Hopf algebras and present its braided and quasitriangular Hopf algebra structure. This generalizes well-known results on $H^{\sigma }$ and $H^R$ type Hopf algebras. Finally, the classification of $H^{\sigma -R}$ type Hopf algebras is given.

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: $H_8\otimes H_4,H_{32,1},H_{32,2},H_{32,3}$. 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.

Let $G$ be a group generated by a set of finite order elements. We prove that any bicrossed product $H_{m,d}\left(q\right)\bowtie k\left[G\right]$ between the generalized Taft algebra $H_{m,d}\left(q\right)$ and group algebra $k\left[G\right]$ is actually the smash product $H_{m,d}\left(q\right)♯k\left[G\right]$. 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 $H_{m,d}\left(q\right)\bowtie k[C_{n_1}\times C_{n_2}]$ is completely presented by generators and relations, where $C_n$ denotes the $n$-cyclic group.