On the Turán Number of $$K_m \vee C_{2k-1}$$

Given a graph H and a positive integer n, the Turán number of H of the order n, denoted by ex(n, H), is the maximum size of a simple graph of order n that does not contain H as a subgraph. Given graphs G and H, \(G \vee H\) denotes the join of G and H. In this paper, we prove \(ex(n, K_m \vee C_{2k-1}) = \left\lfloor \frac{(m+1)n^2}{2(m+2)}\right\rfloor \) for \(n\ge 2(m+2)k-3(m+2)-1\).