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 P ℒ (G) = T r P (G) − P (G) and P (G) = T r P (G) + P (G) , respectively, where P (G) is the path matrix and T r P (G) is the diagonal matrix of the vertex transmissions. The generalized path matrix is P α (G) = α T r P (G) + (1 − α) P (G) , for 0 ≤ α ≤ 1 and ρ 1 α ≥ ρ 2 α ≥ ⋯ ≥ ρ n α are the eigenvalues of P α (G). The generalized path energy can be expressed as E P α (G) = ∑ i = 1 n ρ i α − 2 α P W (G) n , where P W (G) is the path Wiener index of G. We give basic properties of generalized path matrix P α (G). Also, some upper and lower bounds of the generalized path energy of some graphs are studied. [ABSTRACT FROM AUTHOR]