The Number of Blocks of a Graph with Given Minimum Degree

Lei Li,Baoyindureng Wu,A E Matouk

doi:10.1155/2021/6691960

Abstract

A block of a graph is a nonseparable maximal subgraph of the graph. We denote by b G the number of block of a graph G . We show that, for a connected graph G of order n with minimum degree k ≥ 1 , b G < 2 k − 3 / k 2 − k − 1 n . The bound is asymptotically tight. In addition, for a connected cubic graph G of order n ≥ 14 , b G ≤ n / 2 − 2 . The bound is tight.

Highlights

We consider finite, undirected, simple graphs only
Achuthan and Rao [3] determined the maximum number of cut edges in a connected d-regular graph of order p
Let c be a cut vertex lying on an end block Bc

Summary

Introduction

We consider finite, undirected, simple graphs only. Let G (V(G), E(G)) be a graph. e numbers of vertices and edges of G are called the order and the size of G and denoted by v(G) and e(G), respectively. Let G be the graph obtained from identifying each leaf of T with a vertex of a clique of order k + 1 separately. It can be checked that v(Gn) n and b(Gn) (n/2) − 2 for any graph Gn constructed as above

The Proof of Theorem 1

Proof of Theorem 2

Conclusions and Future Work