Workshop on Applied Mathematics and Graph Theory

9th Macedonian Workshop on Applied Mathematics and Graph Theory (2025)

Ohrid, August 6–10, 2025
Book of Abstracts

Invited Talks

A Riviera Model and its Applications – From Rydberg Atoms to Building Codes

Tomislav Došlić
University of Zagreb, Faculty of Civil Engineering, Zagreb, Croatia
doslic@grad.hr

The Riviera model is an analytically tractable simplification of a recent 2D model of urban settlement planning. Despite its simplicity, it exhibits rich behavior and has applications in diverse settings.

In this talk, we review two such applications:

  • Modeling random sequential adsorption with blockade range (relevant to Rydberg atoms),
  • Modeling urban development under conflicting interests (e.g., building codes).

Matrix Orderings and Graphs

Jing Huang
University of Victoria, Canada
huangj@uvic.ca

We study the problem of determining whether a given $0$–$1$ matrix admits row/column orderings that avoid certain forbidden submatrices.

We focus on two key patterns:

  • The $\Gamma$-matrix: rows $(1,1)$ and $(1,0)$,
  • The Slash matrix: rows $(0,1)$ and $(1,0)$.

Matrices avoiding these patterns correspond to adjacency matrices of well-known graph classes:

  • interval graphs,
  • chordal graphs,
  • cocomparability graphs,
  • strongly chordal graphs,
  • chordal bipartite graphs, and more.

These classes admit elegant characterizations via vertex orderings, forbidden induced subgraphs, and polynomial-time recognition algorithms.

Time-Fractional Partial Differential Equations on $\mathbb{R}^d$

Stevan Pilipović
Serbian Academy of Sciences and Arts & University of Novi Sad, Serbia
stevan.pilipovic@gmail.com
(Joint work with Sandro Coriasco and Giovani Girardi)

We study the Cauchy problem:

$$ \partial_t^r u(t, x) + \mathrm{Op}(a)u(t, x) = f(t, x), \quad (t,x) \in (0,\infty) \times \mathbb{R}^d, $$

$$ u(0, x) = u_0(x), $$

where $\partial_t^r$ is the Caputo fractional derivative ($r > 0$), and $\mathrm{Op}(a)$ is a Fourier multiplier or pseudodifferential operator with symbol $a(x,\xi)$ of SG or Hörmander type.

Assuming tempered initial data in suitable distribution spaces, we obtain:

  • A representation formula for the solution,
  • Regularity and decay properties: solutions are smooth and rapidly decreasing in space, modulo continuous-in-time corrections.

Recursive Identity Functions in Computation Theory

Vlado Stankovski
University of Ljubljana, Slovenia
vlado.stankovski@fri.uni-lj.si

This talk explores recursive identity functions in foundational computation models:

  • In lambda calculus, the identity combinator $I = \lambda x.x$ enables self-reference and fixed-point construction.
  • In Turing machines, identity mappings underpin universal simulation.

We extend this to quantum computing, showing how gates like Hadamard, CNOT, and Toffoli implement recursive structures via entanglement and superposition.

The discussion bridges classical recursion, quantum information, and mathematical logic—highlighting identity as a unifying computational primitive.

Contributed Talks

Electrochemical Growth of Graphene and Carbon Nanotubes in Molten Salts: Structural Insights and Data-Guided Optimization

Beti Andonovikj
Faculty of Technology and Metallurgy, UKIM, Skopje, North Macedonia
beti@tmf.ukim.edu.mk
(Joint work with Aleksandar T. Dimitrov and Viktor Andonovikj)

We present advances in electrochemical synthesis of graphene and MWCNTs in molten salts using stationary/pulsed currents.

Key findings:

  • Structural parameters (chirality, wall count, diameters) determined via Raman spectroscopy and Python-based modeling.
  • XRD analysis of graphene layers using both Scherrer equation and a Laue-function-based model for improved accuracy on asymmetric $(002)$ peaks.
  • Interpretable machine learning (decision trees) identifies critical synthesis parameters affecting quality.
  • Enables data-driven optimization of scalable, low-cost nanomaterial production.

Diameter of Nanotori

Vesna Andova
Faculty of Electrical Engineering and Information Technologies, UKIM, Skopje
vesna.andova@gmail.com, vesnaa@feit.ukim.edu.mk
(Joint work with Pavel Dimovski, Martin Knor, and Riste Škrekovski)

A nanotorus $G_{a,b,c}$ is a cubic graph with only hexagonal faces embeddable on a torus, defined by parameters $a,b,c$.

We solve an open problem from Alspach’s survey:

  • If $b \leq a$, then $\mathrm{diam}(G_{a,b,c}) = a$.
  • If $a < b$, we distinguish:
    • $a \leq c < b$,
    • $c < a < b$,
  • and compute the diameter for sufficiently large $b$.

Graph-Directed Fractals: Dimensions and Measures

Jasmina Angelevska Kostadinoska
FEIT, UKIM, Skopje
jasminaa@feit.ukim.edu.mk

Given a finite set of conformal contractions and a $\{0,1\}$-valued adjacency matrix $M$, we define graph-directed self-conformal sets as projections of infinite $M$-admissible sequences.

When $M$ is irreducible:

  • All attractors $F_i$ are quasi self-similar,
  • Share the same Hausdorff dimension,
  • Their $s$-dimensional Hausdorff measures are uniformly comparable.

Results extend to sub-self-conformal sets, generalizing classical IFS theory.

Abelian and Tauberian Type Results for the Fractional Hankel Transform

Sanja Atanasova
FEIT, UKIM, Skopje
ksanja@feit.ukim.edu.mk

We introduce a new Montel space $K_{-1/2}(\mathbb{R}_+)$ tailored for the fractional Hankel transform (FrHT).

Main results:

  • An Abelian theorem: asymptotic behavior of a distribution implies asymptotics of its FrHT.
  • A Tauberian converse: under growth conditions, FrHT asymptotics imply original distribution behavior.

The space ensures equivalence of weak/strong convergence—critical for distributional analysis.

The Hamming Spectrum and Energy of Graphs: Bounds, Factorizations, and Equitable Partitions

Bojana Borovićanin
University of Kragujevac, Serbia
bojanaborovicanin@gmail.com

The Hamming matrix $H(G)$ encodes Hamming distances between binary incidence vectors of vertices.

We establish:

  • Tight eigenvalue bounds for paths $P_n$,
  • Closed-form Hamming spectra for regular graphs, complements, and line graphs,
  • A novel factorization of $\chi_{H(G)}(\lambda)$ in terms of $H(G)$ and $H(L(G))$,
  • A unified framework using equitable partitions to compute spectra for:
    • semi-regular bipartite graphs,
    • complete multipartite graphs,
    • wheels, windmills, coronas.

Distance sequences to bound distance-based topological indices

Peter Dankelmann
University of Johannesburg, South Africa
pdankelmann@uj.ac.za

The distance sequence of a graph is the nondecreasing list of all pairwise distances.

Using its properties, we derive sharp bounds:

  • Harary index: $\displaystyle \sum_{\{u,v\}} \frac{1}{d(u,v)}$ — sharp lower bound for given order/size (solves a problem from Xu–Das–Trinajstić, 2015),
  • Hyper-Wiener index: $\displaystyle \sum_{\{u,v\}} \frac{1}{2}(d(u,v)^2 + d(u,v))$ — sharp upper bound,
  • Extensions to maximal outerplanar graphs, $k$-trees, $k$-connected graphs, and strong products.

On a Class of Mikhlin Multipliers Which Do Not Preserve $L^1$-, $L^\infty$-Regularity and Continuity

Pavel Dimovski
Faculty of Technology and Metallurgy, UKIM, Skopje
dimovski.pavel@gmail.com

We show that any Fourier multiplier with a real-valued, positively homogeneous symbol of order 0—supported in a cone whose dual has nonempty interior, and with sufficiently dominant positive part—fails to preserve:

  • $L^1$-regularity,
  • $L^\infty$-regularity,
  • continuity.

Consequence: no natural wave front set can measure microlocal $L^1$/$L^\infty$/continuity regularity.

Geometric Approach for Topological Indices

Boris Furtula
University of Kragujevac, Serbia
furtula@uni.kg.ac.rs

Treating the pair $(d(u), d(v))$ of end-vertex degrees of an edge $uv$ as a point in $\mathbb{R}^2$ yields geometric degree-based indices.

Starting with the Sombor index:

$$ SO(G) = \sum_{uv \in E(G)} \sqrt{d(u)^2 + d(v)^2}, $$

we overview its descendants:

  • reduced/co-Sombor,
  • elliptic Sombor,
  • Euler Sombor,
  • complementary indices.

These unify chemical intuition with geometric interpretation.

The Role of ChatGPT in Supporting Mathematical Reasoning for Electronics Problems

Branislav Gerazov
FEIT, UKIM, Skopje
gerazov@feit.ukim.edu.mk
(Joint work with Giorgia Nieddu, Bojan Glushica, Maria Cristina Carrisi)

We analyze ChatGPT’s role in undergraduate electronics problem-solving through Brousseau’s Theory of Didactical Situations.

Findings:

  • ChatGPT provides helpful hints but shows inconsistent reasoning,
  • Students’ trust fluctuates due to contradictory outputs,
  • Highlights need for critical validation in AI-assisted learning.

ChatGPT is a double-edged tool: supportive yet requiring careful pedagogical integration.

Fixed Point Theorem for a Class of Chatterjea-Type Mappings

Samoil Malcheski
International Slavic University, Sveti Nikole, Macedonia
samoil.malcheski@msu.edu.mk

We introduce a new class of mappings in complete metric spaces, analogous to Chatterjea mappings (just as Koparde–Waghmode maps generalize Kannan).

We prove:

  • A fixed point theorem,
  • A common fixed point theorem,

for this class, expanding the landscape of contractive-type results.

C-Chain Connected Set in a Topological Space

Zoran Misajleski
Faculty of Civil Engineering, UKIM, Skopje
zokimisajleski@gmail.com

Let $X$ be a topological space and $C \subseteq X$. For a clopen covering $\mathcal{U}$ of $X$, a chain connects $x,y \in X$ if there’s a finite sequence $U_1,\dots,U_k \in \mathcal{U}$ with $x \in U_1$, $y \in U_k$, and $U_i \cap U_{i+1} \ne \emptyset$.

We say $C$ is C-chain connected if for every clopen cover $\mathcal{U}$ and every $x,y \in C$, such a chain exists.

We define related notions and establish their topological properties.

Assessing Pigmented Skin Moles Using Minkowski Fractal Dimensions and Comprehensive Color Channel Analysis

Ilija Mizhimakoski
FEIT, UKIM, Skopje
ilijamizhimakoski@gmail.com
(Joint work with Mia Darkovska, Jasmina Angelevska Kostadinoska, Ana Ristevska Dimitrova, Vesna Andova)

Using the ABCDE criteria, we focus on Color (C) and Border (B) for mole classification.

Methods:

  • Analyze moles in color spaces: HSV, XYZ, YCbCr, Lab,
  • Compute Minkowski fractal dimensions for border irregularity.

Results:

  • Statistically significant differences in color channel features between benign/malignant moles,
  • Color alone is a reliable discriminator,
  • Border analysis adds complementary diagnostic value.

Random Forest for Detecting Thoracic Pathologies in Chest Radiographs

Sijche Pechkova
Faculty of Technology and Metallurgy, UKIM, Skopje
sijche@gmail.com

We propose a Random Forest classifier for thoracic disease detection in chest X-rays.

Advantages over deep learning:

  • Lower computational cost,
  • Better interpretability,
  • Robust performance on imbalanced datasets.

Uses handcrafted features enhanced by Legendre/Chebyshev polynomial expansions to capture subtle pathological patterns.

Support Vector Machines Classifiers for Detecting Thoracic Pathologies in Chest Radiographs

Aleksandar Pechkov
apechkov@gmail.com

We develop an SVM-based method using orthogonal polynomials (Legendre/Chebyshev) as kernel enhancements.

Benefits:

  • Focuses on local pathological regions,
  • Achieves high accuracy with minimal computational overhead,
  • Provides transparent decision rationale—crucial for clinical trust.

Offers a practical, lightweight alternative to black-box deep models.

Spaces of Distributions Having Sobolev Wave Front in a Fixed Conic Set

Bojan Prangoski
Faculty of Mechanical Engineering, UKIM, Skopje
bprangoski@yahoo.com
(Joint work with Stevan Pilipović)

For $r \in \mathbb{R}$, the Sobolev wave front set $WF_r(u)$ captures microlocal $H^r$-regularity.

Given a closed conic set $L \subseteq \Omega \times (\mathbb{R}^n \setminus \{0\})$, define:

$$ \mathcal{D}'_{r,L}(\Omega) = \{ u \in \mathcal{D}'(\Omega) \mid WF_r(u) \subseteq L \}. $$

We:

  • Equip $\mathcal{D}'_{r,L}(\Omega)$ with a locally convex topology,
  • Prove pullback continuity: for smooth $f: \Omega \to U$, the map $f^*: \mathcal{D}'_{r_2,f^*L}(U) \to \mathcal{D}'_{r_1,L}(\Omega)$ is continuous for appropriate $r_1,r_2$,
  • Extend results to smooth manifolds.

Graph-Theoretic Algorithms for Code Equivalence

Simona Samardjiska
Radboud University, Netherlands
simonas@cs.ru.nl

Code equivalence underpins security of post-quantum Fiat-Shamir signatures.

Recent breakthroughs show the best practical algorithms are graph-based:

  • Reduce code equivalence to colored graph isomorphism,
  • Leverage advanced GI solvers (e.g., nauty, Traces).

We survey algorithmic progress and hardness assumptions driving cryptographic design.

Digital Twin and Machine Learning as Tools for Measurement and Diagnostic Methods in Industry 4.0

Mare Srbinovska
FEIT, UKIM, Skopje
mares@feit.ukim.edu.mk

We present a project developing an integrated framework combining:

  • Digital twin modeling,
  • Machine learning for anomaly detection,
  • Real-time sensor data fusion.

Goals:

  • Enable predictive maintenance,
  • Enhance diagnostic precision in industrial systems.

Current phase: simulation modeling and algorithm design. Next: experimental validation on industrial case studies.

Outcomes will include lab setups and educational resources aligned with Industry 5.0 vision.

Graph Pattern-Based Query Rewriting for Fine-Grained Access Control in SPARQL

Riste Stojanov
Faculty of Computer Science, UKIM, Skopje
riste.stojanov@finki.ukim.mk

SPARQL lacks native authorization. We propose policy-based query rewriting for secure RDF querying.

  • Policy: triple $(\Pi, \text{perm}, \rho)$, where $\Pi$ is a quad pattern, $\text{perm} \in \{\text{ALLOW}, \text{DENY}\}$, $\rho$ priority.
  • Rewriting: for query $Q$, compute intersection with policy patterns via variable mapping $\mu$ satisfying:
    1. Structural compatibility,
    2. Type compatibility (URI/literal/blank node).

Implementation:

  • ALLOW: add $\mu(\Pi)$ conjunctively,
  • DENY: subtract via MINUS.

Result: rewritten query returns only authorized data, preserving SPARQL semantics.

The Conflict-free Coloring and its Variations

Riste Škrekovski
FMF-UL & FIŠ Novo mesto, Slovenia
skrekovski@gmail.com

A proper vertex coloring $\phi$ is:

  • Conflict-free if $\forall$ non-isolated $x$, $\exists c$ with $|\phi^{-1}(c) \cap N(x)| = 1$,
  • Odd if $\exists c$ with $|\phi^{-1}(c) \cap N(x)|$ odd.

We summarize results from joint work with M. Petruševski and Y. Caro, and present new findings on degree-choosability of conflict-free colorings (with R. Xu and M. Kashima).

Data-Driven Optimization of Cop Diameter in Ring Spinning Using the Taguchi Design Approach

Emilija Toshikj
Faculty of Technology and Metallurgy, UKIM, Skopje
tosic_emilija@tmf.ukim.edu.mk
(Joint work with Sijche Pechkova)

In ring spinning, cop diameter $B$ is critical for yarn quality.

Using Taguchi method, we optimize:

  • Spindle speed,
  • Traveler mass,
  • Cop height.

Key finding:

  • Doff stage significantly affects cop diameter,
  • Optimal setting: factor levels K1L1M2,
  • Traveler mass and spindle speed have no significant effect.

Validated via S/N ratio analysis for maximum diameter.

The Universe as a Harmonic Oscillator Between Dense Matter and Dense Energy

Kostadin Trenchevski
Faculty of Natural Sciences and Mathematics, UKIM, Skopje
kostadin.trencevski@gmail.com
(Joint work with Ilija Jovceski and Emilija Celakoska)

We model the universe as the intersection of two 4D spheres with radii $\cos(\omega\tau)$ and $\sin(\omega\tau)$.

Features:

  • Time flows only during intersection,
  • Critical mass/energy densities prevent singularities,
  • Introduce “frozen epochs” (no time flow) when density thresholds are met,
  • Dual spatial/temporal Big Bangs with symmetric frozen epochs.

Explains CMB radiation and aligns with JWST observations of early massive galaxies.

Functional Approximations: A Metrical Perspective

Daniel Velinov
Faculty of Civil Engineering, UKIM, Skopje
velinov.daniel@gmail.com

We study approximations of functions $F: \Lambda \times X \to X$ (with $\Lambda \subseteq \mathbb{R}^n$, $X$ Banach) by:

  • Trigonometric polynomials,
  • $\rho$-periodic type functions ($\rho$ a binary relation on $Y$).

We:

  • Define new function spaces,
  • Establish structural and approximation properties,
  • Apply to Volterra integro-differential and PDEs.

QUBOs, Quantum Annealing and Biqbin

Beno Zupanc
Faculty of Information Studies, Novo mesto, Slovenia
zupanc.beno@gmail.com

QUBO (Quadratic Unconstrained Binary Optimization) models hard combinatorial problems and maps to the Ising model—ideal for quantum annealers (e.g., D-Wave).

We compare:

  • Classical simulated annealing (thermal, probabilistic),
  • Quantum annealing (tunneling-based).

Introduce BiqBin, a parallel branch-and-bound solver for Max-Cut and binary quadratic problems.

As part of the QBIQ project, we extended BiqBin into a hybrid classical-quantum solver:

  • Transforms subproblems into QUBOs,
  • Solves them on quantum hardware,
  • Supports custom heuristics.

Publisher: Macedonian Society for Applied Mathematics and Graph Theory
Editor: Vesna Andova
Scientific Committee: Martin Knor, Snježana Majstorovic, Jelena Sedlar, Vesna Andova, Riste Škrekovski, Pavel Dimovski, Beti Andonovikj, Mirko Petruševski
Local Organizing Committee: Vesna Andova, Beti Andonovikj, Pavel Dimovski
Technical Support: Vesna Andova, Beti Andonovikj
Sponsors: UKIM (FEIT, FTM), Faculty of Information Sciences (Novo mesto), M6 Educational Center