On the size of graphs whose cycles have length divisible by a fixed integer

Let G be a simple graph of order n and size m which is not a tree. If 3 is a natural number and the length of every cycle of G is divisible by , then m −2(n − 2), and the equality holds if and only if the following hold: (i) is odd and G is a cycle of order or (ii) is even and G is a generalized θ-graph with paths of length 2 . It is shown that for a (0 mod )-cycle graph, m n < −1 , if is odd, and for a given ε > 0, there exists a (0 mod )-cycle graph G with m n > −1 − ε. Also mn < −2 , if is even, and for a given ε > 0, there exists a (0 mod )-cycle graph G with m n > −2 − ε.