In additive combinatorics, a family of finite sets $A_i$ is said to have bounded doubling if there exists a uniform constant $K$ such that $|A_i + A_i|< K|A_i|$ for all i. In this paper, we study such families in the context of certain symmetric Toeplitz matrices over a field F. In particular, we show that if each matrix has bandwidth b and diagonal entries chosen from a finite set $S \subset F$, then the resulting family admits a doubling constant that depends only on b and the additive properties of $S$, but is independent of the matrix dimension. Also, if the diagonals lie in the image of a fixed-dimensional linear map $L: F^m \to F^{b+1}$, then the doubling constant depends on m rather than b. We include examples to illustrate how one-dimensional constraints on S lead to especially small doubling constants.

In this study, we assume that R is a commutative Noetherian ring with nonzero identity. We present upper bounds for the injective dimension of I, where I is any ideal in the ring R, in terms of the injective dimension of its local cohomology modules and an upper bound for the injective dimension that involves the theory of local cohomology modules. Since I is an ideal in R, we obtain applications of the theory in a general context.

For a left module $_{R}M$ over a non-commutative ring $R$, we define the concept of a strongly semicommutative module as a generalization of the reduced module. This notion constitutes a distinct and stronger category within the class of semicommutative modules. We demonstrate that a module $_{R}M$ is strongly semicommutative if and only if $_{A_{n}(R)}A_{n}(M)$ is strongly semicommutative. Additionally, we establish that $_{R}M$ is strongly semicommutative if and only if $_{R[x]}M[x]$ is strongly semicommutative; this is also equivalent to $_{R[x, x^{-1}]}M[x, x^{-1}]$ being strongly semicommutative. Among our findings, we prove that if $_{R}M$ is strongly semicommutative, then for any reduced submodule $N$ of $M$, the quotient module $M/N$ is also strongly semicommutative. We provide examples of semicommutative modules that are not strongly semicommutative and show that the class of strongly semicommutative modules remains closed under localization.

The non-commuting conjugacy class graph (abbreviated as NCCC-graph) of a finite non-abelian group $H$ is a simple undirected graph whose vertex set is the set of conjugacy classes of non-central elements of $H$ and two vertices, $a^H$ and $b^H$ are adjacent if $a'b' \ne b'a'$ for all $a' \in a^H$ and $b' \in b^H$. In this paper, we compute distance spectrum, distance Laplacian spectrum, distance signless Laplacian spectrum along with their respective energies and Wiener index of NCCC-graphs of $H$ when the central quotient of $H$ is isomorphic to $\mathbb{Z}_p \times \mathbb{Z}_p$ (for any prime $p$) or $D_{2n}$ (for any integer $n \geq 3$). As a consequence, we compute various distance spectra, energies and Wiener index of NCCC-graphs of the dihedral group, dicyclic group, semidihedral group along with the groups $U_{(n,m)}$, $U_{6n}$ and $V_{8n}$. Thus we obtain sequences of positive integers that can be realized as Wiener index of NCCC-graphs of certain groups. In particular, we solve Inverse Wiener index Problem for NCCC-graphs of groups when $n$ is a perfect square. We further characterize the above-mentioned groups such that their NCCC-graphs are D-integral, DL-integral and DQ-integral. We also compare various distance energies of NCCC-graphs of the above mentioned groups and characterize those groups subject to the inequalities involving various distance energies.

The concept of a supercharacter theory for a finite group was introduced in 2008 by Diaconis and Iasaacs in [6]. In their article the notion of irreducible characters and conjugacy classes is generalized to superchacters and superclasses while still maintaining important information about the group. This article continues an investigation of a specific supercharacter theory where the supercharacters are taken to be sums of irreducible characters of the same degree. We show this supercharacter theory construction can be done for all projective special linear groups PSL(2,q) and all special orthogonal groups SO(3,q) where q is any power of an (even or odd) prime.

Let $[n,k,d]_q$ code be a linear code of length $n$, dimension $k$ and minimum Hamming distance $d$ over $GF(q)$. One of the most important problems in coding theory is to construct codes with best possible minimum distances. In this paper 36 new cyclic and quasi-cyclic (QC) codes over GF(11) are presented and the table from [4] is enlarged by adding three new dimensions.

A recursive method is developed for counting domino tilings of a rectangular chessboard (the dimer problem). Based on this method, a new and enhanced recursive algorithm is proposed for solving this problem. Close connections with Fibonacci numbers are traced out.

We construct all nonequivalent (v,k,1) cyclic difference families for 18 sets of parameters v and k for which classification results were not known. We also present the multipliers of all previously classified CDFs with small parameters. Most of the results are double-checked by two different backtrack search algorithms. The usage of an interesting property of the considered objects makes one of these algorithms faster than the other.

The Albertson irregularity measure is defined as $Alb(\Gamma)=\sum_{uv\in E(\Gamma)} \vert d(u)-d(v)\vert.$ In this work, the concept of Albertson energy is extended from simple graphs to graphs with self-loops. Also the expression for the Albertson eigenvalues of a graph with self-loops are given. Some bounds on the Albertson energy of graphs with self-loops and the spread of $Alb(\Gamma_S)$ are obtained. In the last section, the Albertson energy of complete, complete bipartite, crown and thorn graphs with self-loops are computed.

Explicit construction of linear codes over finite fields is one of the most important and challenging problems in coding theory. Due to the centrality of this problem, databases of best-known linear codes (BKLCs) over small finite fields have been available. Recently, new databases for BKLCs over larger alphabets have been introduced. In this work, a new database of BKLCs over the field GF(19) is introduced, containing lower and upper bounds on the minimum distances for codes with lengths up to 150 and dimensions between 3 and 6. Computer searches were conducted on cyclic, constacyclic, quasi-cyclic, and quasi-twisted codes to establish lower bounds. These searches resulted in many new linear codes over GF(19).