arXiv subjects

J. Bagherian

Publications and source records attributed to J. Bagherian.

7 recordsLinked to original sources

arXiv2026

Regular sets of circulant quartic graphs

For a graph $\Gamma=(V,E)$ and nonnegative integers $a$ and $b$, a nonempty proper subset $C \subset V$ is called an $(a,b)$-regular set if every vertex in $C$ has exactly $a$ neighbors in $C$, and every vertex in $V\setminus C$ has exactly $b$ neighbors in $C$. In this paper, we study the existence of such sets in connected Cayley graph $\Gamma = \operatorname{Cay}(\mathbb{Z}_n, S)$. We establish a necessary and sufficient condition for the existence of $(0, |S|)$-regular sets and identify additional conditions under which no such set can exist. We further prove that $(|S|, 0)$-regular sets do not occur in $\Gamma$, and more generally, that no connected Cayley graph $\operatorname{Cay}(G,S)$ contains a $(1, |S|)$-regular set. As a main result, we determine the existence and nonexistence of $(a,b)$-regular sets in connected circulant quartic graphs for all possible values of $a$ and $b$.

math.CO↗

arXiv2026

The Sequence Reconstruction of Permutations under Hamming Metric with Small Errors

The sequence reconstruction problem asks for the recovery of a sequence from multiple noisy copies, where each copy may contain up to $r$ errors. In the case of permutations on $n$ letters under the Hamming metric, this problem is closely related to the parameter $N(n,r)$, the maximum intersection size of two Hamming balls of radius $r$. While previous work has resolved $N(n,r)$ for small radii ($r \leq 4$) and established asymptotic bounds for larger $r$, we present new exact formulas for $r \in \{5,6,7\}$ using group action techniques. In addition, we develop a formula for $N(n,r)$ based on the irreducible characters of the symmetric group $S_n$, along with an algorithm that enables computation of $N(n,r)$ for larger parameters, including cases such as $N(43,8)$ and $N(24,14)$.

math.GR↗

arXiv2022

New Bounds on the Size of Permutation Codes With Minimum Kendall $τ$-distance of Three

We study $P(n,3)$, the size of the largest subset of the set of all permutations $S_n$ with minimum Kendall $τ$-distance $3$. Using a combination of group theory and integer programming, we reduced the upper bound of $P(p,3)$ from $(p-1)!-1$ to $(p-1)!-\lceil\frac{p}{3}\rceil+2\leq (p-1)!-2$ for all primes $p\geq 11$. In special cases where $n$ is equal to $6,7,11,13,14,15$ and $17$ we reduced the upper bound of $P(n,3)$ by $3,3,9,11,1,1$ and $4$, respectively.

math.CO↗

arXiv2020

A conjecture of Cameron and Kiyota on sharp characters with prescribed values

Let $ χ$ be a virtual (generalized) character of a finite group $ G $ and $ L=L(χ)$ be the image of $ χ$ on $ G-\lbrace 1 \rbrace $. The pair $ (G, χ) $ is said to be sharp of type $ L $ if $|G|=\prod _{ l \in L} (χ(1) - l) $. If the principal character of $G$ is not an irreducible constituent of $χ$, the pair $(G,χ)$ is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in $1988$. This conjecture states that if $(G,χ)$ is sharp and $|L|\geq 2$, then the inner product $(χ,χ)_G$ is uniquely determined by $ L $. We then prove that this conjecture is true in the case that $(G,χ) $ is normalized, $χ$ is a character of $ G $, and $ L $ contains at least an irrational value.

math.RT↗

arXiv2009

Burnside-Brauer Theorem and Character Products in Table Algebras

In this paper, we first show that the irreducible characters of a quotient table algebra modulo a normal closed subset can be viewed as the irreducible characters of the table algebra itself. Furthermore, we define the character products for table algebras and give a condition in which the products of two characters are characters. Thereafter, as a main result we state and prove the Burnside-Brauer Theorem on finite groups for table algebras.

math.RT↗

arXiv2008

Standard Character Condition for C-algebras

It is well known that the adjacency algebra of an association scheme has the standard character. In this paper we first define the concept of standard character for C-algebras and we say that a C-algebra has the {\it standard character condition} if it has the standard character. Then we investigate some properties of C-algebras which have the standard character condition and prove that under some conditions a C-algebra has an adjacency algebra homomorphic image. In particular, we obtain a necessary and sufficient condition for which a commutative table algebra comes from an association scheme.

math.RT↗

arXiv2006

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme C over a finite near-field. It is proved that if C is nontrivial, then Aut(C)<AGL(V) where V is the linear space associated with the near-field. In many cases we are able to get more specific information about Aut(C).

math.CO↗