On the Number of cycles in a Graph

TL;DR

This paper derives explicit formulas for counting 7-cycles and paths of lengths 6 and 7 through a specific vertex in a graph, using adjacency matrices and combinatorics.

Contribution

It provides new explicit formulas for counting specific cycles and paths in graphs based on adjacency matrix analysis.

Findings

Formulas for the number of 7-cycles in a graph

Formulas for the number of paths of lengths 6 and 7 through a vertex

Use of adjacency matrices and combinatorics for enumeration

Abstract

In this paper, we obtain explicit formulae for the number of 7-cycles and the total number of paths of lengths 6 and 7 those contain a specific vertex $v_{i}$ in a simple graph G, in terms of the adjacency matrix and with the help of combinatorics.