On the number of spanning trees in graphs with multiple edges

In this paper, we consider a class of nonlinear matrix equation of the type (X+sum _{i=1}^mA_i^{*}X^{-q}A_i-sum _{j=1}^nB_{j}^{*}X^{-r}B_j=Q), where (0 and Q is positive definite. Based on the Schauder fixed point theorem and Bhaskar–Lakshmikantham coupled fixed point theorem, we derive some sufficient conditions for the existence and uniqueness of the positive definite solution to such equations. An iterative method is provided to compute the unique positive definite solution. A perturbation estimation and the explicit expression of Rice condition number of the unique positive definite solution are also established. The theoretical results are illustrated by numerical examples.