arXiv subjects

Omid Khormali

Publications and source records attributed to Omid Khormali.

13 recordsLinked to original sources

arXiv2026

Bipartite Turán Numbers of Trees and Star Forests

The bipartite Turán number of a graph $H$, denoted $\text{ex}(m, n; H)$, is the maximum number of edges in any $H$-free bipartite graph $G = (A, B; E)$ with parts of size $|A| = m$ and $|B| = n$. We study this problem for two families. For a tree $T = T(r, s)$ with parts $R$ and $S$ of sizes $|R| = r \le s = |S|$, we prove \[ (r - 1) n \;\le\; \text{ex}(m, n; T(r, s)) \;\le\; (r - 1) n + O(m) \] for $n$ sufficiently large compared to $m$, $r$, and $s$, determining the leading-order term exactly (with the star case $r=1$ solved with an exact formula). For a star forest $F = \bigcup_{i=1}^k S_{d_i}$ with $d_1 \ge \cdots \ge d_k$, we determine the exact value $\text{ex}(m, n; F) = (k - 1) n + (d_k - 1)(m - k + 1)$ for $n$ sufficiently large, and characterize the unique extremal graph.

math.CO↗

arXiv2026

On the History of the Square and Multiply Algorithm

The square-and-multiply algorithm, also known as binary exponentiation or repeated squaring, is a standard method for fast exponentiation in modern computation. Its historical origins, however, remain uncertain. This paper examines the emergence and progressive formalization of the method through selected primary sources. Particular attention is given to Jamshid al-Kashi's fifteenth-century Miftah al-Hisab, where the procedure is presented explicitly as a general computational method and claimed by al-Kashi as his own innovation. Earlier instances of successive squaring are identified in the works of al-Uqlidisi and al-Biruni, although in these cases the technique appears in particular calculations rather than as a fully articulated general rule. The earliest known antecedent is found in Pingala's prosodic studies in ancient India (c. 200 BCE), which seem to presuppose the conceptual basis of the method in their use of binary representation. As part of the historical development, Legendre's 1798 worked example is one of the earliest documented European use of the algorithm which appears as a subordinate step within a specific number-theoretic computation. The evidence suggests not a single continuous line of transmission, but the repeated independent reappearance of related procedures in distinct contexts. By the twentieth century, square-and-multiply became a special case within the broader theory of addition chains. By exploring this intellectual progression, this paper sheds some light on the historical background of an algorithm that is prominent in modern computation.

math.HO↗

arXiv2026

Topological Data Analysis and Graph-Theoretic Approaches for Tennis Match Prediction

We present two approaches for predicting tennis match outcomes using topological data analysis and graph theory on ATP singles matches from 2000-2025. The first method applies lower-star filtration to player competitive networks, extracting topological features through persistent homology using four summary methods (VAB, HNAV, HWNAV, OW-HNPV) combined with Modified Band Depth analysis. Algorithmic optimizations including ego graph approximations and triangle elimination enable analysis of about 66k matches. Our Random Forest model achieves 66.2% accuracy (AUC = 0.719) using topological, graph-theoretic, and ranking features. Feature importance analysis reveals that rankings contribute 36.3%, centralities 25.5%, and TDA features 24.0%, with topological features providing complementary signal. When rankings are unavailable, the topology-only model maintains 63.56% accuracy, demonstrating that network-derived features alone capture meaningful competitive structure. The second method uses a modified Katz similarity index with temporal edge weighting, achieving 62.48% accuracy on held-out test data. This work represents the first application of lower-star filtration to tennis prediction, provides systematic comparison of four topological summary methods in sports analytics, and demonstrates that TDA can achieve above-chance prediction using network topology alone while providing additional value when combined with traditional features.

cs.LG↗

arXiv2026

Gap-Sums via Quasi-Arithmetic Means with Applications to Fibonacci and Lucas Sequences

We develop a unified framework for studying the integers missing between consecutive terms of an increasing integer sequence, extending Barry's arithmetic gap-sum to geometric and harmonic analogues via the theory of quasi-arithmetic means. All three gap-sums admit a common interpretation: each equals the gap size multiplied by the appropriate mean of the missing integers. Building on this, we prove a general sparse summation theorem expressing the sum of a strictly monotonic function over a sparse integer sequence as the full range sum corrected by the gap-sums of the missing portions. Specializing on the three Pythagorean means recovers a classical formula of al-Kāsh\=ı from the fifteenth century in the arithmetic case, and yields explicit formulas in the geometric and harmonic cases. As a concrete application of the geometric case, we derive a product identity involving the Fuss--Catalan numbers. Applying the harmonic case to the Fibonacci and Lucas sequences, we establish that the harmonic gap-sum converges to $\ln(α)$ exponentially, where $α$ is the golden ratio, and derive explicit two-term asymptotic expansions for the tails of the reciprocal Fibonacci and Lucas series with closed-form coefficients, and establish the asymptotic formula $H_{u_n} \sim n\ln(α)$ for both $u_n = F_n$ and $u_n = L_n$, with explicit $O(1)$ error terms that differ due to their distinct initial conditions. As a further consequence, by comparing the gap-sum expansions with the classical Hardy--Wright expansion of harmonic numbers, we derive exact series identities expressing Euler's constant $γ$ in terms of harmonic numbers at Fibonacci and Lucas indices, and obtain a new identity relating the reciprocal Fibonacci constant $ψ$ and the reciprocal Lucas constant $ψ_L$.

math.NT↗

arXiv2026

Machine Learning-Guided Quota Optimization for Multi-Round Two-Sided Matching

This paper proposes an integrated framework for machine learning-guided quota optimization applied to multi-round sorority recruitment, a small two-sided market where approximately 100 potential new members (PNMs) are matched to three chapters through a structured process governed by the Release Figure Methodology (RFM). Our framework combines a Random Forest classifier trained on historical registration data to generate PNM-chapter compatibility scores, integer linear programs for Round~2 and Round~3 invitation quota optimization balancing fairness, coverage, and efficiency objectives, and a Deferred Acceptance algorithm for final matching. Applied to five years of de-identified recruitment data from a small Midwestern university, and working with only 282 matched training pairs, the compatibility model achieves a cross-validated ROC-AUC of 0.5822, reflecting the inherent difficulty of predicting social compatibility from pre-recruitment registration data in a data-limited setting. Because fairness and coverage constraints dominate quota allocation under noisy scores, the framework is designed to degrade gracefully when ML signal is weak. Optimized quotas closely align with actual coordinator decisions for active chapters, and the Deferred Acceptance algorithm replicates actual 2025 recruitment outcomes with a 96.4% individual-level agreement rate and a 100% match rate across 56 PNMs. An interactive web application implementing the framework is made available to recruitment coordinators. These results support the viability of data-driven approaches to small-market matching with broader applicability to other constrained two-sided markets.

cs.GT↗

arXiv2026

From Historical Puzzles to Grammatical Constraints: Circular Partitions, Generalized Run-Length Encodings, and Polynomial-Time Decidability

Motivated by a historical combinatorial problem that resembles the well-known Josephus problem, we investigate circular partition algorithms and formulate problems in deterministic finite automata with practical algorithms. The historical problem involves arranging individuals on a circle and eliminating every k-th person until a desired group remains. We analyze both removal and non-removal approaches to circular partitioning, establishing conditions for balanced partitions and providing explicit algorithms. We introduce generalized run-length encodings over partitioned alphabets to capture alternating letter patterns, computing their cardinalities using Stirling numbers of the second kind. Connecting these combinatorial structures to formal language theory, we formulate an existence problem: given a context-free grammar over a dictionary and block-pattern constraints on letters, does a valid sentence exist? We prove decidability in polynomial time by showing block languages are regular and applying standard parsing techniques. Complete algorithms with complexity analysis are provided and validated through implementation on both historical and synthetic instances.

cs.FL↗

arXiv2025

On Link-irregular Digraphs

We extend the study of link-irregular graphs to directed graphs (digraphs), where a digraph is link-irregular if no two vertices have isomorphic directed links. We establish that link-irregular digraphs exist on $n$ vertices if and only if $n \geq 5$, and prove that their underlying graphs must contain 3-cycles. We conjecture that link-irregular tournaments exist if and only if $n \geq 6$, providing explicit constructions for $n \leq 8$ and computational verification for $n \leq 100$. We derive lower bounds on the minimum degree and outdegree required for link-irregularity, establish that almost all link-irregular digraphs are nonplanar, and prove that any link-irregular orientable graph admits a link-irregular labeling. Additionally, we construct explicit examples of link-irregular digraphs with constant outdegree and regular tournaments.

math.CO↗

arXiv2025

Hierarchical Persistence Velocity for Network Anomaly Detection: Theory and Applications to Cryptocurrency Markets

We introduce the Overlap-Weighted Hierarchical Normalized Persistence Velocity (OW-HNPV), a novel topological data analysis method for detecting anomalies in time-varying networks. Unlike existing methods that measure cumulative topological presence, we introduce the first velocity-based perspective on persistence diagrams, measuring the rate at which features appear and disappear, automatically downweighting noise through overlap-based weighting. We also prove that OW-HNPV is mathematically stable. It behaves in a controlled, predictable way, even when comparing persistence diagrams from networks with different feature types. Applied to Ethereum transaction networks (May 2017-May 2018), OW-HNPV demonstrates superior performance for cryptocurrency anomaly detection, achieving up to 10.4% AUC gain over baseline models for 7-day price movement predictions. Compared with established methods, including Vector of Averaged Bettis (VAB), persistence landscapes, and persistence images, velocity-based summaries excel at medium- to long-range forecasting (4-7 days), with OW-HNPV providing the most consistent and stable performance across prediction horizons. Our results show that modeling topological velocity is crucial for detecting structural anomalies in dynamic networks.

cs.LG↗

arXiv2025

On Link-irregular labelings of Graphs

We introduce the concept of link-irregular labelings for graphs, extending the notion of link-irregular graphs through edge labeling with positive integers. A labeling is link-irregular if every vertex has a uniquely labeled subgraph induced by its neighbors. We establish necessary and sufficient conditions for the existence of such labelings and define the link-irregular labeling number $η(G)$ as the minimum number of distinct labels required. Our main results include necessary and sufficient conditions for the existence of link-irregular labelings. We show that certain families of graphs, such as bipartite graphs, trees, cycles, hypercubes, and complete multipartite graphs, do not admit link-irregular labelings, while complete graphs and wheel graphs do. Specifically, we prove that $η(K_n) = 2$ for $n \geq 6$ and $η(K_n) = 3$ for $n \in \{3,4,5\}$. For wheel graphs $W_n$, we establish that $η(W_n) \approx \sqrt{2n}$ asymptotically. Finally, we prove that for every positive integer $n$, there exists a graph with a link-irregular labeling number exactly $n$, and provide several results on graph operations that preserve labeling numbers.

math.CO↗

arXiv2025

On the Regularity, Planarity and Edge Bounds of Link-irregular Graphs

A graph $G$ is a link-irregular graph if every two distinct vertices of $G$ have non-isomorphic links. The link of a vertex $v$ in $G$ is the subgraph induced by the neighbors of $v$ in $G$. Ali, Chartrand and Zhang [Discussiones Mathematicae. Graph Theory, 45(1) (2025) p.95] conjectured that there exists no regular link-irregular graph. In this paper, we show that the existence of an $r$-regular link irregular graph is very likely for large enough $r$. In particular, we provide a 7-regular link irregular graph on 12 vertices, which serves as a counterexample to the conjecture. Additionally, we prove that no bipartite link-irregular graphs exist, and there are no regular link-irregular graphs on $n$-vertices for $n \leq 9$. Also, we determine upper and lower bounds for the number of edges of link-irregular graphs. Furthermore, we show the minimum number of edges in a link-irregular graph on the $n$ vertices is $Ω(n\sqrt{\log n})$. Finally, we prove that all but finitely many link-irregular graphs are non-planar, and there is no regular link-irregular planar graphs.

math.CO↗

arXiv2023

A fast topological approach for predicting anomalies in time-varying graphs

Large time-varying graphs are increasingly common in financial, social and biological settings. Feature extraction that efficiently encodes the complex structure of sparse, multi-layered, dynamic graphs presents computational and methodological challenges. In the past decade, a persistence diagram (PD) from topological data analysis (TDA) has become a popular descriptor of shape of data with a well-defined distance between points. However, applications of TDA to graphs, where there is no intrinsic concept of distance between the nodes, remain largely unexplored. This paper addresses this gap in the literature by introducing a computationally efficient framework to extract shape information from graph data. Our framework has two main steps: first, we compute a PD using the so-called lower-star filtration which utilizes quantitative node attributes, and then vectorize it by averaging the associated Betti function over successive scale values on a one-dimensional grid. Our approach avoids embedding a graph into a metric space and has stability properties against input noise. In simulation studies, we show that the proposed vector summary leads to improved change point detection rate in time-varying graphs. In a real data application, our approach provides up to 22% gain in anomalous price prediction for the Ethereum cryptocurrency transaction networks.

cs.LG↗

arXiv2020

Turán numbers for hypergraph star forests

Fix a graph $F$. We say that a graph is {\it $F$-free} if it does not contain $F$ as a subgraph. The {\it Turán number} of $F$, denoted $\mathrm{ex}(n,F)$, is the maximum number of edges possible in an $n$-vertex $F$-free graph. The study of Turán numbers is a central problem in graph theory. The goal of this paper is to generalize a theorem of Lidický, Liu and Palmer [{\it Electron.\ J.\ of Combin.}\ {\bf 20} (2016)] that determines $\mathrm{ex}(n,F)$ for $F$ a forest of stars. In particular, we consider generalizations of the problem to three different well-studied hypergraph settings and in each case we prove an asymptotic result for all reasonable parameters defining our "star forests".

math.CO↗

arXiv2016

Regular colorings and factors of regular graphs

An $(r-1,1)$-coloring of an $r$-regular graph $G$ is an edge coloring such that each vertex is incident to $r-1$ edges of one color and $1$ edge of a different color. In this paper, we completely characterize all $4$-regular pseudographs (graphs that may contain parallel edges and loops) which do not have a $(3,1)$-coloring. An $\{r-1,1\}$-factor of an $r$-regular graph is a spanning subgraph in which each vertex has degree either $r-1$ or $1$. We prove various conditions that that must hold for any vertex-minimal $5$-regular pseudographs without $(4,1)$-colorings or without $\{4,1\}$-factors. Finally, for each $r\geq 6$ we construct graphs that are not $(r-1,1)$-colorable and, more generally, are not $(r-t,t)$-colorable for small $t$.

math.CO↗