Estimating the Zagreb indices and the spectral radius of triangle- and quadrangle-free connected graphs

The first ( M 1 ) and second ( M 2 ) Zagreb indices and the spectral radius ( ρ ) are topological indices. This Letter presents upper bounds for these indices of triangle- and quadrangle-free connected graphs, in terms of the number of vertices (n), number of edges (m), and the radius (r). These bounds are: M 1 ( G ) ⩽ n ( n + 1 - r ) , M 2 ( G ) ⩽ m ( n + 1 - r ) , ρ ⩽ n + 1 - r , and we determine the graphs for which the bounds are attained.