Some years ago, in D’Hoker and Phong (1989) studied the functional determinants of Laplacian on Mandelstam diagrams. They considered some renormalizations of the functional determinants of Laplacian on Mandelstam diagrams and explored their applications in String Theory. Recently, on quite a different subject, in Qing (1997) studied the renormalized energy for Ginzburg-Landau vortices on closed...

Let [Formula: see text] be a simple graph with vertices, edges having Laplacian eigenvalues text]. The energy LE[Formula: is defined as text], where the average degree of Radenković and Gutman conjectured that among all trees order path has smallest energy. family diameter In this paper, we show any tree greater than thereby proving conjecture for We also truth number non-pendent vertices at mo...

A graph whose edges are labeled either as positive or negative is called a signed graph. In this article, we extend the notion of composition of (unsigned) graphs (also called lexicographic product) to signed graphs. We employ Kronecker product of matrices to express the adjacency matrix of this product of two signed graphs and hence find its eigenvalues when the second graph under composition ...

developing electrical energy is one of the basic needs for development, so equipping the rural regions and villiages is one of the main responisiblities of thirdworld countries. in this study, we used economic evaluation indexes to compare electrification to baghtaj village by power grid expansion and application of solar photovoltaic systems. economic evaluation indexes suggests that in the ab...