The concept of geometric-arithmetic indices was introduced in the chemical graph theory. These indices are defined by the following general formula:     ( ) 2 ( ) uv E G u v u v Q Q Q Q GA G , where Qu is some quantity that in a unique manner can be associated with the vertex u of graph G. In this paper the exact formula for two types of geometric-arithmetic index of Vphenylenic nanotube ar...

Computing the exact ideal and nadir criterion values is a very ‎important subject in ‎multi-‎objective linear programming (MOLP) ‎problems‎‎. In fact‎, ‎these values define the ideal and nadir points as lower and ‎upper bounds on the nondominated points‎. ‎Whereas determining the ‎ideal point is an easy work‎, ‎because it is equivalent to optimize a ‎convex function (linear function) over a con...

This note is part of implementation of a program in foundations of mathematics to find exact versions of all unprovability theorems known so far, a program initiated by A. Weiermann. In this note we find the exact version of unprovability of the graph minor theorem restricted to planar graphs and some lower and upper bounds in the general case of all graphs. 1 Unlabelled growth constants Let us...

Recent measurements of cosmic-ray electron and positron fluxes by PAMELA and ATIC experiments may indicate the existence of annihilating dark matter with large annihilation cross section. We discuss its possible relation to other astrophysical/cosmological observations : gamma-rays, neutrinos, and big-bang nucleosynthesis. It is shown that they give stringent constraints on some annihilating da...

Let G be a digraph with n vertices and let A(G) be its adjacency matrix. A monic polynomial f (x) of degree at most n is called an annihilating polynomial of G if f (A(G)) = 0. G is said to be annihilatingly unique if it possesses a unique annihilating polynomial. Difans and diwheels are two classes of annihilatingly unique digraphs. In this paper, it is shown that the complete product of difan...