All Research Papers Published In Utilitas Mathematica

Research Article
10.61091/um126-09

Partial hyperbolicity and average pseudo-orbit tracing property

Mar 13, 2026
Utilitas Mathematica
Manseob Lee

In this paper, given a homeomorphism f of a compact metric space X, we show that the set of all asymptotic average shadowable points of f is an open and invariant set and f has the asymptotic average shadowing property if and only if the set of all asymptotic average shadowable points of f is X if and only if any Borel probability measure μ of X has the asymptotic average shadowing property.

Research Article
10.61091/um126-08

Supermagic labeling of \(C_n\Box C_m\)

Feb 12, 2026
Utilitas Mathematica
Dalibor Froncek

A supermagic labeling (often also called vertex-magic edge labeling) of a graph \(G(V,E)\) with \(|E|=q\) is a bijection from \(E\) to the set of first \(k\) positive integers such that the sum of labels of all incident edges of every vertex \(x\in V\) is equal to the same integer \(c\). An existence of a supermagic labeling of Cartesian product of two cycles, \(C_{n}\Box C_m\) for \(n,m\geq4\) and both \(n,m\) even and for any \(C_n\Box C_n\) with \(n\geq3\) was proved by Ivančo. Ivančo also conjectured that such labeling is possible for any \(C_n\Box C_m\) with \(n,m\geq3\). We prove his conjecture for all \(n,m\) odd that are not relatively prime.

Research Article
10.61091/um126-05

Ergodic results in operator algebras

Jan 27, 2026
Utilitas Mathematica
Francesco Fidaleo

We provide a hierarchy of “nonconventional ergodic theorems” in quantum setting involving operators and unitaries acting on the Hilbert space of the Gelfand-Naimark-Segal covariant representation associated to a reference C*-dynamical system. The first two levels correspond to the Mean Ergodic Theorem by J. von Neumann involving a unitary, and the ergodic theorem by Kovács and Szúcs relative to unitarily implemented automorphisms of von Neumann algebras, respectively. As a consequence, we provide multiple correlation results concerning the ergodic behaviour of “three-operator” expectations.

Research Article
10.61091/um126-02

Double domination number of inflated graphs

Jan 27, 2026
Utilitas Mathematica
Magima M + 1 more +1

In a graph, a vertex is said to dominate itself and all its neighbors. A subset \(S\subseteq V(G)\) is a double dominating set of a graph \(G\), if every vertex of \(V\) is dominated by at least two vertices of \(S\). The double domination number denoted by \(\gamma_{2\times(G)}\) is the minimum cardinality of a double dominating set. Graph operations are fundamental in graph theory and have various applications across different fields including network analysis, parallel computing and electrical circuit design. This paper studies the problem of finding the double domination number under unary and binary operations of graphs. We investigate the double domination number of graphs under unary operations such as inflation and cubic inflation. Also, we introduced two new unary operations inspired from inflation operation and studied the impact of these operations on double domination number. Further, we explore the double domination number of edge corona and neighborhood corona of two graphs. Additionally, we study the double domination number of various corona operations of two graphs combined with subdivision of a graph and \(R-\)graph.

Journal Issue
10.61091/um126

Jan 27, 2026
Utilitas Mathematica

Research Article
10.61091/um125-09

Spectral radius and the 2-power of Hamilton paths

Dec 29, 2025
Utilitas Mathematica
Zia Ullah Khan

We determine the maximum number of edges of a graph without containing the 2-power of a Hamilton path. Using this result, we establish a spectral condition for a graph containing the 2-power of a Hamilton path. Furthermore, we characterized the extremal graphs with the largest spectral radius that do not contain the 2-power of a Hamilton path.

Research Article
10.61091/um125-02

Encoding and enumerating acyclic orientations of graphs

Dec 29, 2025
Utilitas Mathematica
Walter Carballosa + 2 more +2

In this work we study the acyclic orientations of graphs. We obtain an encoding of the acyclic orientations of the complete \(p\)-partite graph with size of its parts \(n_1,n_2,\ldots,n_p\) via a vector with \(p\) symbols and length \(n=n_1+n_2+\ldots+n_p\) when the parts are fixed but not the vertices in each part. We also give a recursive way to construct all acyclic orientations of a complete multipartite graph, this construction can be done by computer easily in order \(\mathcal{O}(n)\). Furthermore, we obtain a closed formula for non-isomorphic acyclic orientations of both the complete multipartite graphs and the complete multipartite graphs with a directed spanning tree. Moreover, we obtain a closed formula for the number of acyclic orientations of a complete multipartite graph \(K_{n_1,\ldots,n_p}\) with labelled vertices. Finally, we obtain a way encode all acyclic orientations of an arbitrary graph as a permutation code. Using the codification mentioned above we obtain sharp upper and lower bounds of the number of acyclic orientations of a graph.

Research Article
10.61091/um124-12

On local antimagic chromatic number of the join of two special families of graphs | II

Sep 28, 2025
Utilitas Mathematica
Gee-Choon Lau + 1 more +1

Null graphs (respectively, 1-regular graphs) are the only regular graphs with local antimagic chromatic number 1 (respectively, undefined). In this paper, we proved that the join of 1-regular graph and a null graph has local antimagic chromatic number 3. As a by-product, we also obtained many families of (possibly disconnected or regular) bipartite and tripartite graph with local antimagic chromatic number 3.

Research Article
10.61091/um124-06

Directed graphs of integers with arcs determined by an arithmetic function

Sep 28, 2025
Utilitas Mathematica
Phakhinkon Napp Phunphayap + 2 more +2

We introduce a new arc in directed graphs of integers. Our arcs are determined by the values of the popular arithmetic functions such as the divisor function \(\tau\), the prime divisors counting functions \(\omega\) and \(\Omega\), and the sum of digits function \(s_b\), evaluated at the multiples \(N\) of a particular integer. Among other things, we determine the positive integers that have arcs to all except a finite number of positive integers. We also propose some possible research problems at the end of this article.

Research Article
1
10.61091/um123-14

Certified domination critical graphs upon vertex removal

Jun 26, 2025
Utilitas Mathematica
Mateusz Miotk

A set \(D\) of vertices of a graph \(G=(V_G,E_G)\) is a dominating set of \(G\) if every vertex in \(V_G-D\) is adjacent to at least one vertex in \(D\). The domination number of a graph \(G\), denoted by \(\gamma(G)\), is the cardinality of a smallest dominating set of \(G\). A subset \(D\subseteq V_G\) is called a certified dominating set of \(G\) if \(D\) is a dominating set of \(G\), and every vertex in \(D\) has either zero or at least two neighbours in \(V_G-D\). The cardinality of a smallest certified dominating set of \(G\) is called the certified domination number of \(G\), and it is denoted by \(\gamma_{\rm cer}(G)\). A vertex \(v\) of \(G\) is certified critical if \(\gamma_{\rm cer}(G -v) < \gamma_{\rm cer}(G)\), and a graph \(G\) is vertex certified domination critical or \(\gamma_{cer}\)-critical if the removal of any vertex reduces its certified domination number. In this paper, we give examples and properties of certified critical vertices and vertex certified domination critical graphs. As an example of an application of certified critical vertices, we give a constructive characterisation of trees for which the smaller partite set is a minimum certified dominating set.