arXiv subjects

Abdulaziz M. Alanazi

Publications and source records attributed to Abdulaziz M. Alanazi.

10 recordsLinked to original sources

arXiv2026

Some Comments on Regular Overpartitions modulo $2^k$

For coprime integers $\ell,μ\ge 2$, Alanazi, Munagi, and Saikia (2026) studied $\overline{R}_{\ell,μ}(n)$, the number of overpartitions of $n$ in which no part is divisible by $\ell$ or by $μ$, together with the single-modulus analogue $\overline{R}_{\ell (n)$. We record a simple combinatorial mechanism that determines both functions modulo every power of $2$ in terms of the number of distinct part sizes of the underlying ordinary partition. We also deduce a clean characterization of $\overline{R}_{\ell}(n)$ and $\overline{R}_{\ell,μ}(n)$ modulo $4$ in terms of perfect squares.

math.NT↗

arXiv2025

On the Diminished Sombor Index of Fixed-Order Molecular Graphs With Cyclomatic Number at Least 3

For a graph $G$ with edge set $E$, let $d(u)$ denote the degree of a vertex $u$ in $G$. The diminished Sombor (DSO) index of $G$ is defined as $DSO(G)=\sum_{uv\in E}\sqrt{(d(u))^2+(d(v))^2}(d(u)+d(v))^{-1}$. The cyclomatic number of a graph is the smallest number of edges whose removal makes the graph acyclic. A connected graph of maximum degree at most $4$ is known as a molecular graph. The primary motivation of the present study comes from a conjecture concerning the minimum DSO index of fixed-order connected graphs with cyclomatic number $3$, posed in the recent paper [F. Movahedi, I. Gutman, I. Redžepović, B. Furtula, Diminished Sombor index, MATCH Commun. Comput. Chem. 95 (2026) 141--162]. The present paper gives all graphs minimizing the DSO index among all molecular graphs of order $n$ with cyclomatic number $\ell$, provided that $n\ge 2(\ell-1)\ge4$.

math.CO↗

arXiv2025

Signed Partitions and Rogers-Ramanujan type Identities

George Andrews [\emph{Bull. Amer. Math. Soc.}, 2007, 561--573] introduced the idea of a \emph{signed partiton} of an integer; similar to an ordinary integer partitions, but where some of the parts could be negative. Further, Andrews reinterpreted the classical Göllnitz--Gordon partition identities in terms of signed partitions. In the present work, we provide interpretations of the sum sides of Rogers--Ramanujan type identities, including a new signed partition interpretation of the Göllnitz--Gordon identities, different from that of Andrews. Both analytic and bijective proofs are presented.

math.CO↗

arXiv2025

On a Conjecture Concerning the Complementary Second Zagreb Index

The complementary second Zagreb index of a graph $G$ is defined as $cM_2(G)=\sum_{uv\in E(G)}|(d_u(G))^2-(d_v(G))^2|$, where $d_u(G)$ denotes the degree of a vertex $u$ in $G$ and $E(G)$ represents the edge set of $G$. Let $G^*$ be a graph having the maximum value of $cM_2$ among all connected graphs of order $n$. Furtula and Oz [MATCH Commun. Math. Comput. Chem. 93 (2025) 247--263] conjectured that $G^*$ is the join $K_k+\overline{K}_{n-k}$ of the complete graph $K_k$ of order $k$ and the complement $\overline{K}_{n-k}$ of the complete graph $K_{n-k}$ such that the inequality $k<\lceil n/2 \rceil$ holds. We prove that (i) the maximum degree of $G^*$ is $n-1$ and (ii) no two vertices of minimum degree in $G^*$ are adjacent; both of these results support the aforementioned conjecture. We also prove that the number of vertices of maximum degree in $G^*$, say $k$, is at most $-\frac{2}{3}n+\frac{3}{2}+\frac{1}{6}\sqrt{52n^2-132n+81}$, which implies that $k<5352n/10000$. Furthermore, we establish results that support the conjecture under consideration for certain bidegreed and tridegreed graphs. In the aforesaid paper, it was also mentioned that determining the $k$ as a function of the $n$ is far from being an easy task; we obtain the values of $k$ for $5\le n\le 149$ in the case of certain bidegreed graphs by using computer software and found that the resulting sequence of the values of $k$ does not exist in "The On-Line Encyclopedia of Integer Sequences" (an online database of integer sequences).

math.CO↗

arXiv2024

Some Properties of Overpartitions into Nonmultiples of Two Integers

We consider properties of overpartitions that are simultaneously {\ell}-regular and μ-regular, where {\ell} and μ are positive relatively prime integers. We prove a seven-way combinatorial identity related to these overpartitions. We also prove several congruence properties satisfied by this class of partitions (and a further related class) using both generating functions and modular forms with Radu's Algorithm.

math.NT↗

arXiv2024

Extrinsic n-type semiconductor transition in ZrSe2 with the metallic character through hafnium substitution

Two dimensional layered materials exhibit versatile electronic properties in their different phases. The intrinsic electronic properties of these materials can be modulated through doping or intercalation. In this study, we investigated the electronic properties of Hf doped ZrSe2 single crystals using angle-resolved photoemission spectroscopy (ARPES) combined with first principles density functional theory (DFT) calculations. It is observed that the valence band maxima of ZrSe2, located below the Fermi level, undergo a significant change with the introduction of Hf substitution. Hf can introduce extra charges into the conduction band, rather than making a mixed structure of HfSe2 and ZrSe2 band structure, which can cross the Fermi level. Compared to the semiconducting band structure of ZrSe2, we observed that the conduction band crosses the Fermi level at the high symmetry M point in Hf-doped ZrSe2. This suggests an increase of electron type carriers around the Fermi level, resulting in an extrinsic charge carrier density in the conduction band, which can form a metallic behaviour. It can be noticed that the Hf cations can create disorder in the form of excess atoms of Zr, which yields more carriers in the conduction band in the shape of smeared bands. The tails of the smeared band occupied the d-orbitals extended into the Fermi level and left the d band below. Similarly, the electrical resistance measurements further confirm the metallic-like character of Hf doped ZrSe2 compared to the semiconductor ZrSe2, indicating increased carriers. This metallic like behavior is suggested to be predisposed by the extrinsic electrons induced by the substitutional disorder. This study further demonstrates the possibility of band gap engineering through heavy metal doping in 2D materials.

cond-mat.mtrl-sci↗

arXiv2023

Harmonic-Arithmetic Index of (Molecular) Trees

Let $G$ be a graph. Denote by $d_x$, $E(G)$, and $D(G)$ the degree of a vertex $x$ in $G$, the set of edges of $G$, and the degree set of $G$, respectively. This paper proposes to investigate (both from mathematical and applications points of view) those graph invariants of the form $\sum_{uv\in E(G)}φ(d_v,d_w)$ in which $φ$ can be defined either using well-known means of $d_v$ and $d_w$ (for example: arithmetic, geometric, harmonic, quadratic, and cubic means) or by applying a basic arithmetic operation (addition, subtraction, multiplication, and division) on any of two such means, provided that $φ$ is a non-negative and symmetric function defined on the Cartesian square of $D(G)$. Many existing well-known graph invariants can be defined in this way; however, there are many exceptions too. One of such uninvestigated graph invariants is the harmonic-arithmetic (HA) index, which is obtained from the aforementioned setting by taking $φ$ as the ratio of the harmonic and arithmetic means of $d_v$ and $d_w$. A molecular tree is a tree whose maximum degree does not exceed four. Given the class of all (molecular) trees with a fixed order, graphs that have the largest or least value of the HA index are completely characterized in this paper.

math.CO↗

arXiv2022

On the Maximum Sigma Index of k-Cyclic Graphs

Let $G$ be a graph with edge set $E(G)$. Denote by $d_w$ the degree of a vertex $w$ of $G$. The sigma index of $G$ is defined as $\sum_{uv\in E(G)}(d_u-d_v)^2$. A connected graph of order $n$ and size $n+k-1$ is known as a connected $k$-cyclic graph. Abdo, Dimitrov, and Gutman [Discrete Appl. Math. 250 (2018) 57-64] characterized the graphs having the greatest sigma index over the family of all connected graphs of a fixed order. The primary goal of the present note is to determine graphs possessing the greatest sigma index from the class of all connected $k$-cyclic graphs of a fixed order.

math.CO↗

arXiv2021

On Andrews' partitions with parts separated by parity

In this paper, we present a generalization of one of the theorems in [G. E. Andrews, Partitions with parts separated by parity, \textit{Annals of Combinatorics} \textbf{23}(2019), 241 - 248], and give its bijective proof. Further variations of related partition functions are studied resulting in a number of interesting identities.

math.NT↗

arXiv2019

Power Partitions and Semi-m-Fibonacci Partitions

George Andrews recently proved a new identity between the cardinalities of the set of Semi-Fibonacci partitions and the set of partitions into powers of two with all parts appearing an odd number of times. This paper extends the identity to the set of Semi-$m$-Fibonacci partitions of $n$ and the set of partitions of $n$ into powers of $m$ in which all parts appear with multiplicity not divisible by $m$.

math.CO↗