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The first and the second Zagreb index of a graph, usually defined as the sum of the squares of degrees over all vertices and the sum of the products of degrees of edge endvertices over all edges, respectively, are tightly related to the numbers of walks of length two and three in the graph. We provide here a lower bound on the spectral radius of adjacency matrix A of graph in terms of its Zagreb indices, based on the properties of the least square approximation of the vector A 2 j with the vectors Aj and j , where j is the all-one vector. The bound is sharp for all graphs with two main eigenvalues, surpassing the range of sharpness of other bounds among connected graphs. [ABSTRACT FROM AUTHOR]

In this paper, we determine the unique maximum (or minimum) extremal graph for general spectral radius, zeroth-order general Randić index, general sum-connectivity index, general Platt index, second Zagreb index and multiplicative Zagreb indices in the class of cacti with n ≥ 4 vertices and k ≥ 0 cycles. By applying our new result, we demonstrate the unique maximum extremal graph for general spectral radius, zeroth-order general Randić index, general sum-connectivity index, general Platt index, second Zagreb index and second multiplicative Zagreb index in the class of cacti with n ≥ 4 vertices. Furthermore, we also determine the unified maximum extremal graphs for general spectral radius, zeroth-order general Randić index and the second multiplicative Zagreb index in the class of cacti with n ≥ 4 vertices and k ≥ 1 pendant vertices. [ABSTRACT FROM AUTHOR]