arXiv subjects

Julian Allagan

Publications and source records attributed to Julian Allagan.

8 recordsLinked to original sources

arXiv2026

Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization

We study an operator $\Theta$ on finite integer sequences, where $\Theta(\sigma)_i$ counts the entries to the left of $\sigma_i$ that are strictly smaller than $\sigma_i$. This operator is a variant of the so-called Lehmer code. For every sequence $\sigma$, the image $\Theta(\sigma)$ is an inversion sequence, and the restriction of $\Theta$ to permutations of $[0,n-1]$ is a bijection onto inversion sequences of length $n$. We characterize the fixed points of $\Theta$ by avoidance of the pattern $101$ together with a saturation condition, prove that they are counted by the Catalan numbers, and give an explicit recursive bijection with Dyck paths. We also show that the sequences whose first $\Theta$-image is fixed are precisely those avoiding both $101$ and $201$. Finally, we prove finite stabilization for all inversion sequences, exhibit a family attaining the maximal stabilization time, and show that the second stabilization level is not closed under classical patterns.

math.CO↗

arXiv2026

Robustness, Cost, and Attack-Surface Concentration in Phishing Detection

Phishing detectors built on engineered website features attain near-perfect accuracy under i.i.d.\ evaluation, yet deployment security depends on robustness to post-deployment feature manipulation. We study this gap through a cost-aware evasion framework that models discrete, monotone feature edits under explicit attacker budgets. Three diagnostics are introduced: minimal evasion cost (MEC), the evasion survival rate $S(B)$, and the robustness concentration index (RCI). On the UCI Phishing Websites benchmark (11\,055 instances, 30 ternary features), Logistic Regression, Random Forests, Gradient Boosted Trees, and XGBoost all achieve $\mathrm{AUC}\ge 0.979$ under static evaluation. Under budgeted sanitization-style evasion, robustness converges across architectures: the median MEC equals 2 with full features, and over 80\% of successful minimal-cost evasions concentrate on three low-cost surface features. Feature restriction improves robustness only when it removes all dominant low-cost transitions. Under strict cost schedules, infrastructure-leaning feature sets exhibit 17-19\% infeasible mass for ensemble models, while the median MEC among evadable instances remains unchanged. We formalize this convergence: if a positive fraction of correctly detected phishing instances admit evasion through a single feature transition of minimal cost $c_{\min}$, no classifier can raise the corresponding MEC quantile above $c_{\min}$ without modifying the feature representation or cost model. Adversarial robustness in phishing detection is governed by feature economics rather than model complexity.

cs.LG↗

arXiv2026

Four Dominion Growth Regimes in Trees: Forcing, Fibonacci Enumeration, Periodicity, and Stability

We study the dominion zeta(G), defined as the number of minimum dominating sets of a graph G, and analyze how local forcing and boundary effects control the flexibility of optimal domination in trees. For path-based pendant constructions, we identify a sharp forcing threshold: attaching a single pendant vertex to each path vertex yields complete independence with zeta = 2^gamma, whereas attaching two or more pendant vertices forces a unique minimum dominating set. Between these extremes, sparse pendant patterns produce intermediate behavior: removing endpoint pendants gives zeta = 2^(gamma - 2), while alternating pendant attachments induce Fibonacci growth zeta asymptotic to phi^gamma, where phi is the golden ratio. For complete binary trees T_h, we establish a rigid period-3 law zeta(T_h) in {1, 3} despite exponential growth in |V(T_h)|. We further prove a sharp stability bound under leaf deletions, zeta(T_h - X) <= 2^{m_1(X)} zeta(T_h), where m_1(X) counts parents that lose exactly one child; in particular, deleting a single leaf preserves the domination number and exactly doubles the dominion.

math.CO↗

arXiv2026

Exact Dominion of the Prism Graph: Enumeration by Congruence Class via Cyclic Words

Let G_n = C_n square P_2 denote the prism (circular ladder) graph on 2n vertices. By encoding column configurations as cyclic words, domination is reduced to local Boolean constraints on adjacent factors. This framework yields explicit formulas for the dominion zeta(G_n), stratified by n mod 4, with the exceptional cases n in {3, 6} confirmed computationally. Together with the known domination numbers gamma(G_n), these results expose distinct arithmetic regimes governing optimal domination, ranging from rigid forcing to substantial enumerative flexibility, and motivate quantitative parameters for assessing structural robustness in parametric graph families.

math.CO↗

arXiv2025

Dominion of some graphs

Given a graph G equals (V,E), a subset S subset of V is a dominating set if every vertex in V minus S is adjacent to some vertex in S. The dominating set with the least cardinality, gamma, is called a gamma-set which is commonly known as a minimum dominating set. The dominion of a graph G, denoted by zeta(G), is the number of its gamma-sets. Some relations between these two seemingly distinct parameters are established. In particular, we present the dominions of paths, some cycles and the join of any two graphs.

math.CO↗

arXiv2025

Statistical and Machine Learning Analysis of Traffic Accidents on US 158 in Currituck County: A Comparison with HSM Predictions

This study extends previous hotspot and Chi-Square analysis by Sawyer \cite{sawyer2025hotspot} by integrating advanced statistical analysis, machine learning, and spatial modeling techniques to analyze five years (2019--2023) of traffic accident data from an 8.4-mile stretch of US 158 in Currituck County, NC. Building upon foundational statistical work, we apply Kernel Density Estimation (KDE), Negative Binomial Regression, Random Forest classification, and Highway Safety Manual (HSM) Safety Performance Function (SPF) comparisons to identify comprehensive temporal and spatial crash patterns. A Random Forest classifier predicts injury severity with 67\% accuracy, outperforming HSM SPF. Spatial clustering is confirmed via Moran's I test ($I = 0.32$, $p < 0.001$), and KDE analysis reveals hotspots near major intersections, validating and extending earlier hotspot identification methods. These results support targeted interventions to improve traffic safety on this vital transportation corridor. Our objective is to provide actionable insights for improving safety on US 158 while contributing to the broader understanding of rural highway safety analysis through methodological advancement beyond basic statistical techniques.

cs.LG↗

arXiv2025

Strong Central 2-Trees with Tail Degrees {2, 3}: Structural Characterization and Uniqueness Criteria

We study strong $r$-central $2$-trees whose non-central vertices have degrees in $\{2,3\}$, focusing on the cases $r=1,2,3$. For each $r$, we derive exact degree constraints relating the maximum degree $Δ$ to the numbers of degree-$3$ and degree-$2$ tail vertices. In the unicentral case ($r=1$), we prove that the fan graph is the unique realization for all $n\ge 3$. For bicentral $2$-trees ($r=2$), we show that the number of degree-$3$ vertices is always even, establish sharp uniqueness results for $x\in\{0,2\}$, prove existence for all feasible values of $Δ$, and obtain linear lower bounds on the number of non-isomorphic realizations. For tricentral $2$-trees ($r=3$), we characterize extremal configurations, establish a divisibility constraint on the tail parameters, and prove a quadratic lower bound on the number of non-isomorphic graphs for infinitely many values of $n$. These results provide a unified structural framework for central $2$-trees with bounded tail degrees and highlight sharp transitions between rigidity and combinatorial growth.

math.CO↗

arXiv2025

Golden Ratio Growth and Phase Transitions in Chromatic Counts of Circular Chord Graphs

We study generalized circular chord graphs $\mathcal C^{(k)}_n$, formed from a cycle $C_n$ by adding fixed-offset chords of length $k$ and, for even $n$, diameters. Using transfer matrix methods, we derive exact formulas for 3-colorings when $k=3$: for odd $n$, we obtain \[ P(\mathcal{C}_n^{(3)},3) = L_n + 2\cos\left(\frac{2πn}{3}\right) + 2s_n + 2 \] where $L_n$ is the Lucas sequence and $(s_n)$ satisfies $s_{n+3} = -s_{n+2} - s_n$, yielding golden-ratio asymptotic growth $φ^n + O(ρ^n)$ along odd indices. For even $n$, we construct a paired-window transfer matrix that exactly enumerates $P(\mathcal{C}_{2m}^{(3)},3)$ while capturing diameter constraints. The chromatic counts exhibit pronounced modular patterns across residue classes without universal vanishing rules (see OEIS A383733). We provide efficient algorithms for exact enumeration and demonstrate applications to cyclic scheduling problems where these results serve as feasibility engines for airline gate assignment, wireless sensor networks, and multiprocessor task coordination.

math.CO↗