arXiv subjects

Mostafa Einollahzadeh

Publications and source records attributed to Mostafa Einollahzadeh.

4 recordsLinked to original sources

arXiv2023

Minimum trace norm of real symmetric and Hermitian matrices with zero diagonal

We obtain tight lower bounds for the trace norm $\Vert \cdot \Vert_1$ of some matrices with diagonal zero, in terms of the entry-wise $L^1$-norm (denoted by $\Vert \cdot \Vert_{(1)}$). It is shown that on the space of nonzero real symmetric matrices $A$ of order $n$ with diagonal zero, the minimum value of the quantity $\frac{\Vert A\Vert_1}{\Vert A\Vert_{(1)}}$ is equal to $\frac{2}{n}$. The answer of the similar problem in the space of Hermitian matrices, is also obtained to be equal to $\tan(\fracπ{2n})$. The equivalent "dual" form of these results, give some upper bounds for the distance to the nearest diagonal matrix for a given symmetric or Hermitian matrix, when the distance is computed in the spectral norm.

math.SP↗

arXiv2021

Proof of a conjecture on the algebraic connectivity of a graph and its complement

For a graph $G$, let $λ_2(G)$ denote its second smallest Laplacian eigenvalue. It was conjectured that $λ_2(G) + λ_2(\overline{G}) \geq 1$, where $\bar{G}$ is the complement of $G$. Here, we prove this conjecture in the general case. Also, we will show that $\max\{λ_2(G), λ_2(\overline{G})\} \geq 1 - O(n^{-\frac 13})$, where $n$ is the number of vertices of $G$.

math.CO↗

arXiv2019

Proof of a Conjecture on the Seidel Energy of Graphs

Let $G$ be a graph with the vertex set $ \lbrace v_1,\ldots,v_n \rbrace$. The Seidel matrix of $G$ is an $n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $ v_{i} $ and $ v_{j} $ are adjacent and otherwise is $ 1 $. The Seidel energy of $G$ is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix of $G$. Haemers conjectured that the Seidel energy of any graph of order $n$ is at least $2n-2$ and, up to Seidel equivalence, the equality holds for $ K_{n} $. We establish the validity of Haemers' Conjecture in general.

math.CO↗

arXiv2016

Tannakian formalism for fiber functors over tensor categories

In this paper we generalize Tannakian formalism to fiber functors over general tensor categories. We will show that (under some technical conditions) if the fiber functor has a section, then the source category is equivalent to the category of comodules over a Hopf algebra in the target category. We will also give a description of this Hopf algebra using the notion of framed objects.

math.CT↗