International Journal of Applied Mathematical Research, 3 (2) (2014) 178-189 c Science Publishing Corporation www.sciencepubco.com/index.php/IJAMR doi: 10.14419/ijamr.v3i2.2409 Research Paper On the number of paths of lengths 3 and 4 in a graph Nazanin Movarraei * , M. M. Shikare Department of Mathematics, University of Pune, Pune 411 007 (India) *Corresponding author E-mail: nazanin.movarraei@gmail.com Copyright c 2014 Nazanin Movarraei, M. M. Shikare. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we obtain explicit formulae for the total number of paths of lengths 3 and 4 in a simple graph G. We also determine some formulae for the number of paths of lengths 3 and 4 each of which starts from an specific vertex v i and for the number of v i - v j paths of lengths 3 and 4 in a simple graph G, in terms of the adjacency matrix and with the helps of combinatorics. Keywords : Adjacency Matrix, Cycle, Graph Theory, Path, Subgraph, Walk . 1. Introduction In a simple graph G, a walk is a sequence of vertices and edges of the form v 0 ,e 1 ,v 1 , ..., e k ,v k such that the edge e i has ends v i-1 and v i . A walk is called closed if v 0 = v k . If the vertices of a walk are distinct then that walk is called a path and a cycle is a non-trivial closed path. It is known that if a graph G has adjacency matrix A=[a ij ], then for k = 0, 1, ... , the ij-entry of A k is the number of v i - v j walks of length k in G. It is also known that tr(A n ) is the sum of the diagonal entries of A n and d i is the degree of the vertex v i . In 1971, Frank Harary and Bennet Manvel [1], gave a formula for the number of triangles in simple graphs as given by the following theorem: Theorem 1.1 If G is a simple graph with adjacency matrix A, then the number of 3-cycles in G is 1 6 tr(A 3 ). (It is known that tr(A 3 )= n X i=1 a (3) ii = X i6=j a (2) ij a ij ). They also gave formulae for the number of cycles of lengths 4 and 5 in simple graphs. Their proofs are based on the following fact: The number of n-cycles (n= 3, 4, 5) in a graph G is equal to 1 2n (tr(A n )- x) where x is the number of closed walks of length n, which are not n-cycles. In 1986, Tomescu [2], gave some formulae for the number of paths of length s having k edges in common with a fixed s-path of a complete graph. In 1994, Bax [3], gave an algorithm to count number of all paths and v i - v j paths in a graph. His algorithm was about counting number of all paths in a graph and it can not count number of paths of an specific size.