Let G be a graph with n vertices and let 1; 2; : : : ; n be the eigenvalues of Randic matrix. The Randic Estrada index of G is REE(G) = Ón i=1 ei . In this paper, we establish lower and upper bounds for Randic index in terms of graph invariants such as the number of vertices and eigenvalues of graphs and improve some previously published lower bounds.

Abstract The modified Bessel function (MBF) of the first kind is a fundamental special function in mathematics with applications in a large number of areas. When the order of this function is integer, it has an integral representation which includes the exponential of the cosine function. Here we generalize this MBF to include a fractional parameter, such that the exponential in the previously mentioned integral is replaced by a Mittag-Leffler function. The necessity for this generalization arises from a problem of communication in networks. We find the power series representation of the fractional MBF of the first kind, as well as some differential properties. We give some examples of its utility in graph/networks analysis and mention some fundamental open problems for further investigation.