Utilitas Mathematica

ISSN: 0315-3681 (print)

Utilitas Mathematica is a historical journal that focuses on sharing research in statistical designs and combinatorial mathematics. It has been publishing since 1972. From 2024 onward, it publishes four volumes per year in March, June, September and December. Utilitas Mathematica has gained recognition and visibility in the academic community and is indexed in renowned databases such as MathSciNet, Zentralblatt, and Scopus. The scope of the journal includes; graph theory, design theory, extremal combinatorics, enumeration, algebraic combinatorics, combinatorial optimization, discrete geometry, convex geometry, Ramsey theory, automorphism groups, coding theory, finite geometries, chemical graph theory, etc.

Andrew Bowling1, Bryan Freyberg2
1Department of Chemical Engineering, University of Minnesota Duluth, MN 55812 USA
2Department of Mathematics and Statistics, University of Minnesota Duluth, MN 55812 USA
Abstract:

Let \(G=(V,E,F)\) be a planar graph with vertex set \(V\), edge set \(E\), and set of faces \(F.\) For nonnegative integers \(a,b,\) and \(c\), a type \((a,b,c)\) face-magic labeling of \(G\) is an assignment of \(a\) labels to each vertex, \(b\) labels to each edge, and \(c\) labels to each face from the set of integer labels \(\{1,2,\dots a|V|+b|E|+c|F|\}\) such that each label is used exactly once, and for each \(s\)-sided face \(f \in F,\) the sum of the label of \(f\) with the labels of the vertices and edges incident with \(f\) is equal to some fixed constant \(\mu_s\) for every \(s.\) We find necessary and sufficient conditions for every quadruple \((a,b,c,n)\) such that the \(n\)-prism graph \(Y_n \cong K_2 \square C_n\) admits a face-magic labeling of type \((a,b,c)\).

S. Madhumitha1, S. Naduvath1
1Department of Mathematics Christ University, Bangalore, India
Abstract:

A special type of algebraic intersection graph called the \(n\)-inordinate invariant intersection graph has been constructed based on the symmetric group, and its structural properties are studied in the literature. In this article, we discuss the different types of dominator coloring schemes of the \(n\)-inordinate invariant intersection graphs and their complements, \(n\)-inordinate invariant non-intersection graphs, by obtaining the required coloring pattern and determining the graph invariant associated with the coloring.

Oleksiy Dovgoshey1,2
1Department of Theory of Functions, Institute of Applied Mathematics and Mechanics of NASU, Slovyansk, Ukraine
2Department of Mathematics and Statistics, University of Turku, Turku, Finland
Abstract:

Let \(G\) be a connected graph and let \(d_G\) be the geodesic distance on \(V(G)\). The metric spaces \((V(G), d_{G})\) were characterized up to isometry for all finite connected \(G\) by David C. Kay and Gary Chartrand in 1965. The main result of this paper expands this characterization on infinite connected graphs. We also prove that every metric space with integer distances between its points admits an isometric embedding in \((V(G), d_G)\) for suitable \(G\).

Brian Hopkins1, Jesús Sistos Barrón2, Hua Wang3
1Department of Mathematics and Statistics, Saint Peter’s University, Jersey City NJ 07306 USA
2Department of Mathematics, University of Georgia, Athens GA 30602 USA
3Department of Mathematical Sciences, Georgia Southern University, Statesboro GA 30458 USA
Abstract:

MacMahon extensively studied integer compositions, including the notion of conjugation. More recently, Agarwal introduced \(n\)-color compositions and their cyclic versions were considered by Gibson, Gray, and Wang. In this paper, we develop and study a conjugation rule for cyclic \(n\)-color compositions. Also, for fixed \(\ell\), we identify and enumerate the subset of self-conjugate compositions of \(\ell\), as well as establish a bijection between these and the set of cyclic regular compositions of \(\ell\) with only odd parts.

A. Lourdusamy1, T. Mathivanan2
1Department of Mathematics, St. Xavier’s College (Autonomous), Palayamkottai – 627 002, Tamilnadu, India
2Department of Mathematics, Athoor Cooperative Arts and Science College, Seeval Saragu, Dindigul – 624 303, Tamilnadu, India
Abstract:

The covering cover pebbling number, \(\sigma(G)\), of a graph \(G\), is the smallest number such that some distribution \(D \in \mathscr{K}\) is reachable from every distribution starting with \(\sigma(G)\) (or more) pebbles on \(G\), where \(\mathscr{K}\) is a set of covering distributions. In this paper, we determine the covering cover pebbling number for two families of graphs those do not contain any cycles.

Mohammed Alshammari1, Sergey Kitaev1, Chaoliang Tang2, Tianyi Tao2, Junchi Zhang2
1Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, United Kingdom
2Shanghai Center for Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai 200433, China
Abstract:

Jeff Remmel introduced the concept of a \(\mathit{k}\)-11-representable graph in 2017. This concept was first explored by Cheon et al. in 2019, who considered it as a natural extension of word-representable graphs, which are exactly 0-11-representable graphs. A graph \(G\) is \(k\)-11-representable if it can be represented by a word \(w\) such that for any edge (resp., non-edge) \(xy\) in \(G\) the subsequence of \(w\) formed by \(x\) and \(y\) contains at most \(k\) (resp., at least \(k+1\)) pairs of consecutive equal letters. A remarkable result of Cheon et al. is that  any graph is 2-11-representable, while it is still unknown whether every graph is 1-11-representable. Cheon et al. showed that the class of 1-11-representable graphs is strictly larger than that of word-representable graphs, and they introduced a useful toolbox to study 1-11-representable graphs, which was extended by additional powerful tools suggested by Futorny et al. in 2024. In this paper, we prove that all graphs on at most 8 vertices are 1-11-representable hence extending the known fact that all graphs on at most 7 vertices are 1-11-representable. Also, we discuss applications of our main result in the study of multi-1-11-representation of graphs we introduce in this paper analogously to the notion of multi-word-representation of graphs suggested by Kenkireth and Malhotra in 2023.

Harsha Vardhan K S1, Anuradha D S1, Jaganathan B1
1Department of Computer Science, Vellore Institute of Technology, Chennai, India
Abstract:

Topological Indices (TIs) are quantitative measures derived from molecular geometry and are utilized to predict physicochemical properties. Although more than 3000 TIs have been documented in the published literature, only a limited number of TIs have been effectively employed owing to certain limitations. A significant drawback is the higher degeneracy resulting from the lower discriminative power. TIs utilize simple graphs in which atoms and bonds are conceptualized as the vertices and edges of mathematical graphs. As multiple edges are not supported in these graphs, double and triple bonds are considered single. Consequently, the molecular structure undergoes alterations during the conversion process, which ultimately affects the discriminative power. In this investigation, indices for double-bond incorporation were formulated to preserve structural integrity. This study addresses, demonstrates, and verifies a set of double-bonded indices. The indices demonstrated promising results, exhibiting enhanced discriminative power when validated for polycyclic aromatic hydrocarbons using regression analysis. These indices and their potential applications will significantly contribute to QSAR/QSPR studies.

Linlin Cui1, Feng Li1
1Computer College, Qinghai Normal University, Xi’ning, 810000, Qinghai, China
Abstract:

With the rapid development of wireless communication networks, it brings more and more convenience to users. However, with the expansion of network size, the limitation of channel resources in network communication is becoming more obvious. Effective channel assignment has a great impact on the quality of communication networks. However, in real communication networks, underutilization of channels and excessive number of channels produce large interference, so it is necessary to find a reasonable channel assignment method. In this paper, we study the optimal channel assignment strategy for the Cartesian product of an \(m\)-vertex complete bipartite graph and an \(m\)-order cycle, where \(m\geq 5\) is odd. Determines the exact value and lower bound of its radio number.

Abaid ur Rehman Virk1, Iftikhar Ahmed2, Murat Cancan3
1Department of Mathematics, University of Management and Technology, Lahore, Pakistan
2University of Agriculture, Faisalabad, Burewala Campus, Pakistan
3Faculty of Education,Yuzuncu Yil University, Van, Turkey
Abstract:

This study introduces a novel approach to investigating Sombor indices and applying machine learning methods to assess the similarity of non-steroidal anti-inflammatory drugs (NSAIDs). The research aims to predict the structural similarities of nine commonly prescribed NSAIDs using a machine learning technique, specifically a linear regression model. Initially, Sombor indices are calculated for nine different NSAID drugs, providing numerical representations of their molecular structures. These indices are then used as features in a linear regression model trained to predict the similarity values of drug combinations. The model’s prediction performance is evaluated by comparing the predicted similarity values with the actual similarity values. Python programming is employed to verify accuracy and conduct error analysis.

Leila Vusuqi1, Adel P. Kazemi1, Farshad Kazemnejad2
1Faculty of Mathematical Sciences, University of Mohaghegh Ardabili P.O.\ Box 5619911367, Ardabil, Iran
2Department of Mathematics, Faculty of Basic Sciences, Ilam University P.O.Box 69315-516, Ilam, Iran
Abstract:

Total dominator total coloring of a graph is a total coloring of the graph such that each object of the graph is adjacent or incident to every object of some color class. The minimum namber of the color classes of a total dominator total coloring of a graph is called the total dominator total chromatic number of the graph. Here, we will find the total dominator chromatic numbers of wheels, complete bipartite graphs and complete graphs.

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Call for papers

Special issue: Dynamical systems and differential equations in applied sciences
Guest editors: Renhai Wang, Mirelson Martins Freitas, Nguyen Huy Tuan
Submission deadline: 03 January 2026

Special Issues

The Combinatorial Press Editorial Office routinely extends invitations to scholars for the guest editing of Special Issues, focusing on topics of interest to the scientific community.