Percolation critical polynomial as a graph invariant

The critical exponent: a novel graph invariant

A surprising result of FitzGerald and Horn (1977) shows that A◦α := (aα ij) is positive semidefinite (p.s.d.) for every entrywise nonnegative n× n p.s.d. matrix A = (aij) if and only if α is a positive integer or α ≥ n− 2. Given a graph G, we consider the refined problem of characterizing the setHG of entrywise powers preserving positivity for matrices with a zero pattern encoded by G. Using al...

On Szeged Polynomial of a Graph

Conformally invariant scaling limits in planar critical percolation

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov’s theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLEκ), a one-parameter family of conformally invaria...

On the Roots of Hosoya Polynomial of a Graph

Let G = (V, E) be a simple graph. Hosoya polynomial of G is d(u,v) H(G, x) = {u,v}V(G)x , where, d(u ,v) denotes the distance between vertices u and v. As is the case with other graph polynomials, such as chromatic, independence and domination polynomial, it is natural to study the roots of Hosoya polynomial of a graph. In this paper we study the roots of Hosoya polynomials of some specific g...

on szeged polynomial of a graph