arXiv subjects

Idan Eisner

Publications and source records attributed to Idan Eisner.

5 recordsLinked to original sources

arXiv2026

A New Lower Bound on the Spectral Radius of Graphs with Prescribed Average Degree

This work establishes an improved lower bound for the spectral radius of a graph given its average degree. The new bound follows from an exact solution of the fractional relaxation of the problem. Our findings lead to an affirmative answer to a conjecture by Hong (1993) for graphs with specific average degrees -- as the extremal graphs that meet our bound are proven to have a minimal and maximal degree that differ by at most one. Furthermore, we provide an exact characterization of the conditions that permit such discrete realizations. We prove that for a fixed number of vertices $n$, the number of valid edge configurations grows at least linearly with $n$, achieving an average asymptotic order of $\Theta(n\log n)$.

math.CO↗

arXiv2024

Entropy and the growth rate of universal covering trees

This work studies the relation between two graph parameters, $\rho$ and $\Lambda$. For an undirected graph $G$, $\rho(G)$ is the growth rate of its universal covering tree, while $\Lambda(G)$ is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW). It is well known that $\rho(G) \geq \Lambda(G)$ for all graphs, and that graphs with $\rho=\Lambda$ exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length $\ell$ NBRW is $O(1)$ if $\rho = \Lambda$ and $\Omega(\ell)$ if $\rho > \Lambda$. As a consequence we exhibit infinitely many non-trivial examples of graphs with $\rho = \Lambda$.

math.CO↗

arXiv2016

Exotic Cluster Structures on $SL_n$ with Belavin-Drinfeld Data of Minimal Size, I. The Structure

Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin-Drinfeld classification of solutions to the classical Yang Baxter equation. For any non trivial Belavin-Drinfeld data of minimal size for $SL_{n}$, we give an algorithm for constructing an initial seed $Σ$ in $\mathcal{O}(SL_{n})$. The cluster structure $\mathcal{C}=\mathcal{C}(Σ)$ is then proved to be compatible with the Poisson bracket associated with that Belavin-Drinfeld data, and the seed $Σ$ is locally regular. This is the first of two papers, and the second one proves the rest of the conjecture: the upper cluster algebra $\bar{\mathcal{A}}_{\mathbb{C}}(\mathcal{C})$ is naturally isomorphic to $\mathcal{O}(SL_{n})$, and the correspondence of Belavin-Drinfeld classes and cluster structures is one to one.

math.QA↗

arXiv2015

Exotic cluster structures on $SL_n$ with Belavin-Drinfeld data of minimal size: II. Correspondence between cluster structures an BD triples

Using the notion of compatibility between Poisson brackets and cluster structures in the coordinate rings of simple Lie groups, Gekhtman Shapiro and Vainshtein conjectured a correspondence between the two. Poisson Lie groups are classified by the Belavin--Drinfeld classification of solutions to the classical Yang Baxter equation. For any non trivial Belavin--Drinfeld data of minimal size for $SL_{n}$, the companion paper constructed a cluster structure with a locally regular initial seed, which was proved to be compatible with the Poisson bracket associated with that Belavin--Drinfeld data. This paper proves the rest of the conjecture: the corresponding upper cluster algebra $\overline{\mathcal{A}}_{\mathbb{C}}(\mathcal{C})$ is naturally isomorphic to $\mathcal{O}\left(SL_{n}\right)$, the torus determined by the BD triple generates theaction of $(\mathbb{C}^{*})^{2k_{T}}$ on $\mathbb{C}\left(SL_{n}\right)$, and the correspondence between Belavin--Drinfeld classes and cluster structures is one to one.

math.QA↗

arXiv2014

Exotic cluster structures on $SL_{5}$

A conjecture by Gekhtman, Shapiro and Vainshtein suggests a correspondence between the Belavin - Drinfeld classification of solutions of the classical Yang - Baxter equation and cluster structures on simple Lie groups. This paper confirms the conjecture for $SL_{5}$. Given a Belavin - Drinfeld class, we construct the corresponding cluster structure in $\mathcal{O}\left(SL_{5}\right)$, and show that it satisfies all parts of the conjecture.

math.QA↗