On a Linear Combination of Zagreb Indices

The modified first Zagreb connection index of a triangle-free and quadrangle-free graph G is equal to 2 M 2 ( G ) − M 1 ( G ) , where M 2 ( G ) and M 1 ( G ) are the well-known second and first Zagreb indices of G , respectively. This paper involves the study of the linear combination 2 M 2 ( G ) − M 1 ( G ) of M 2 ( G ) and M 1 ( G ) when G is a connected graph of a given order and cyclomatic number. More precisely, graphs having the minimum value of the graph invariant 2 M 2 − M 1 are determined from the class of all connected graphs of order n and cyclomatic number c y , when c y ≥ 1 and n ≥ 2( c y − 1) .