Spin Chains in Combinatorics

A note on polyomino chains with extremum general sum-connectivity index

The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $chi_{alpha}$ values from a certain collection of polyomino ...

Gapped and gapless phases of frustration-free spin-1/2 chains

We consider a family of translation-invariant quantum spin chains with nearestneighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψ be an arbitrary twoqubit state. We consider a chain of n qubits with open boundary conditions and Hamiltonian Hn(ψ) which is defined as the sum of rank-1 projectors onto...

Individual Displacements in Hashing with Coalesced Chains

We study the asymptotic distribution of the displacements in hashing with coalesced chains, for both late-insertion and early-insertion. Asymptotic formulas for means and variances follow. The method uses Poissonization and some stochastic calculus.

Spin Glasses: A Challenge for Mathematicians - Cavity and Mean Field Models, cby Michel Talagrand

Combinatorics and distributions of partial injections

We obtain several combinatorial results about chains, cycles and orbits of the elements of the symmetric inverse semigroup ISn and the set Tn of nilpotent elements in ISn. We also get some estimates for the growth of |ISn| and |Tn|, and study random products of elements from ISn.