arXiv subjects

M. Khatami

Publications and source records attributed to M. Khatami.

4 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 $\tau$-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 $\tau$-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 $ \chi $ be a virtual (generalized) character of a finite group $ G $ and $ L=L(\chi)$ be the image of $ \chi $ on $ G-\lbrace 1 \rbrace $. The pair $ (G, \chi) $ is said to be sharp of type $ L $ if $|G|=\prod _{ l \in L} (\chi(1) - l) $. If the principal character of $G$ is not an irreducible constituent of $\chi$, the pair $(G,\chi)$ 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 $(G,\chi)$ is sharp and $|L|\geq 2$, then the inner product $(\chi,\chi)_G$ is uniquely determined by $ L $. We then prove that this conjecture is true in the case that $(G,\chi) $ is normalized, $\chi$ is a character of $ G $, and $ L $ contains at least an irrational value.

math.RT↗