Doubly Stochastic Matrices and Modified Laplacian Matrices of Graphs

We consider modified Laplacian matrices of graphs, obtained by adding the identity matrix to the Laplacian matrix $L_G$ of a graph $G$. This results in a positive definite matrix $\tilde{L}_G$. The inverse of $\tilde{L}_G$ is a doubly stochastic matrix. The goal of this paper is to investigate this inverse matrix and how it depends on properties of the underlying graph $G$. In particular, we introduce a general monotonicity property for the entries of the inverse, and derive a sharper version for the case of path graphs. Finally, we show that, in the case of a path graph, the entries of the inverse can be expressed in terms of Fibonacci numbers via an $LU$ factorization. We also establish a lower bound for the diagonal entries of this inverse for a tree as a function of the distances between vertices. Furthermore, we present a simple and efficient algorithm for computing the inverse when the graph is a tree. Moreover, for a general graph, we show that the diagonal entries of this inverse is strictly largest in each row and column. Finally, we discuss a connection to partial differential equations, such as the heat equation.