The generalized path matrix and energy

We define the path Laplacian matrix and the path signless Laplacian matrix of a simple connected graph G as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is the path matrix and [Formula: see text] is the diagonal matrix of the vertex transmissions. The generalized path matrix is [Formula: see text], for [Formula: see text] and [Formula: see text] are the eigenvalues of [Formula: see text]. The generalized path energy can be expressed as [Formula: see text], where [Formula: see text] is the path Wiener index of G. We give basic properties of generalized path matrix [Formula: see text]. Also, some upper and lower bounds of the generalized path energy of some graphs are studied.