We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect matchings of a certain graph GT,γ that is constructed from the surface by recursive glueing of elementary pieces that we call tiles. We also give a second formula...

Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of sl3, which allows us to give an explicit formula for the colored Jones polynomial of th...

We show that the A-polynomial A(L,M) of a 2-bridge knot b(p, q) is irreducible if p is prime, and if (p − 1)/2 is also prime and q , 1 then the L-degree of A(L,M) is (p−1)/2. This shows that the AJ conjecture relating the A-polynomial and the colored Jones polynomial holds true for these knots, according to work of the second author. We also study relationships between the A-polynomial of a 2-b...

We consider a committee voting setting in which each voter approves of a subset of candidates and based on the approvals, a target number of candidates are selected. Aziz et al. (2015) proposed two representation axioms called justified representation and extended justified representation. Whereas the former can be tested as well as achieved in polynomial time, the latter property is coNP-compl...

In 2022, the idea of a bivariate polynomial which represents number complete subgraphs graph with corresponding common neighborhood systems has been introduced in [3]. present work, we extend this notion to more restricted case by considering maximal connected cliques given graph. Besides characterizing corona two graphs, establish clique resulting from graphs. Received: June 2, 2023Accepted: J...

let $g$ be a simple graph of order $n$‎. ‎we consider the‎ ‎independence polynomial and the domination polynomial of a graph‎ ‎$g$‎. ‎the value of a graph polynomial at a specific point can give‎ ‎sometimes a very surprising information about the structure of the‎ ‎graph‎. ‎in this paper we investigate independence and domination‎ ‎polynomial at $-1$ and $1$‎.