Graphs with maximum size and lower bounded girth

For integers n ≥ 4 and ν ≥ n + 1 , let e x ( ν ; { C 3 , … , C n } ) denote the maximum number of edges in a graph of order ν and girth at least n + 1 . The { C 3 , … , C n } -free graphs with order ν and size e x ( ν ; { C 3 , … , C n } ) are called extremal graphs and denoted by E X ( ν ; { C 3 , … , C n } ) . We prove that given an integer k ≥ 0 , for each n ≥ 2 log 2 ( k + 2 ) there exist extremal graphs with ν vertices, ν + k edges and minimum degree 1 or 2. Considering this idea we construct four infinite families of extremal graphs. We also see that minimal ( r ; g ) -cages are the exclusive elements in E X ( ν 0 ( r , g ) ; { C 3 , … , C g − 1 } ) .