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1.
On metric dimensions of hypercubes
Aleksander Kelenc, Aoden Teo Masa Toshi, Riste Škrekovski, Ismael G. Yero, 2023, izvirni znanstveni članek

Opis: In this note we show two unexpected results concerning the metric, the edge metric and the mixed metric dimensions of hypercube graphs. First, we show that the metric and the edge metric dimensions of ▫$Q_d$▫ differ by at most one for every integer ▫$d$▫. In particular, if ▫$d$▫ is odd, then the metric and the edge metric dimensions of ▫$Q_d$▫ are equal. Second, we prove that the metric and the mixed metric dimensions of the hypercube ▫$Q_d$▫ are equal for every ▫$d \ge 3$▫. We conclude the paper by conjecturing that all these three types of metric dimensions of ▫$Q_d$▫ are equal when d is large enough.
Ključne besede: edge metric dimension, mixed metric dimension, metric dimension, hypercubes
Objavljeno v DKUM: 21.05.2024; Ogledov: 2; Prenosov: 0
.pdf Celotno besedilo (255,81 KB)
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2.
Dimensional deviations in Ti-6Al-4V discs produced with different process parameters during selective laser melting
Snehashis Pal, Marijana Milković, Riad Ramadani, Janez Gotlih, Nenad Gubeljak, Radovan Hudak, Igor Drstvenšek, Matjaž Finšgar, Tomaž Brajlih, 2023, izvirni znanstveni članek

Opis: When manufacturing complicated products where both material and design play a role, especially thin and curved components, it is difcult to maintain accurate dimensions in Selective Laser Melting. Considering these difculties, this article presents the dimensional errors in the fabrication of Ti-6Al-4V discs and their thermomechanics during manufacturing. Various combinations of laser processing parameters were used to fabricate the 2.00 mm thick discs with a diameter of 5.70 mm. It was found that the thickness shortened and the round shape changed to an oval shape for most of the discs. The thickness decreased along the build-up direction from the bottom to the top and formed a taper that increased with increasing energy density (ED). The horizontal diameter of the discs changed slightly, while the vertical diameters changed remarkably with increasing ED. On the other hand, reducing the laser power resulted in a reduction of the roundness error, while it caused a reduction of the thickness. The hatch spacing signifcantly afected the volume of the melt pool and caused a change in the vertical diameter. The central part of the curved surface of the discs became concave and the concavity increased due to the increasing ED.
Ključne besede: dimension, Ti-6Al-4V, lase power, scanning speed, hatch spacing, selective laser melting
Objavljeno v DKUM: 29.03.2024; Ogledov: 94; Prenosov: 2
.pdf Celotno besedilo (4,48 MB)
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3.
Incidence dimension and 2-packing number in graphs
Dragana Božović, Aleksander Kelenc, Iztok Peterin, Ismael G. Yero, 2022, izvirni znanstveni članek

Opis: Let ▫$G=(V,E)$▫ be a graph. A set of vertices ▫$A$▫ is an incidence generator for ▫$G$▫ if for any two distinct edges ▫$e,f \in E(G)$▫ there exists a vertex from ▫$A$▫ which is an endpoint of either ▫$e$▫ or ▫$f$▫. The smallest cardinality of an incidence generator for ▫$G$▫ is called the incidence dimension and is denoted by ▫$\dim_I(G)$▫. A set of vertices ▫$P \subseteq V(G)$▫ is a 2-packing of ▫$G$▫ if the distance in ▫$G$▫ between any pair of distinct vertices from ▫$P$▫ is larger than two. The largest cardinality of a 2-packing of ▫$G$▫ is the packing number of ▫$G$▫ and is denoted by ▫$\rho(G)$▫. In this article, the incidence dimension is introduced and studied. The given results show a close relationship between ▫$\dim_I(G)$▫ and ▫$\rho(G)$▫. We first note that the complement of any 2-packing in graph ▫$G$▫ is an incidence generator for ▫$G$▫, and further show that either ▫$\dim_I(G)=|V(G)|-\rho(G)$▫ or ▫$\dim_I(G)=|V(G)-|\rho(G)-1$▫ for any graph ▫$G$▫. In addition, we present some bounds for ▫$\dim_I(G)$▫ and prove that the problem of determining the incidence dimension of a graph is NP-hard.
Ključne besede: incidence dimension, incidence generator, 2-packing
Objavljeno v DKUM: 18.08.2023; Ogledov: 234; Prenosov: 19
.pdf Celotno besedilo (434,03 KB)
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4.
Distance-based Invariants and Measures in Graphs
Aleksander Kelenc, 2019, doktorska disertacija

Opis: This doctoral dissertation is concerned with aspects on distance related topics in graphs. We study three main topics, namely a recently introduced measure called the Hausdorff distance of graphs and two new graph invariants - the edge metric dimension and the mixed metric dimension of graphs. All three topics are part of the metric graph theory since they are tightly connected with the basic concept of distance between two vertices of a graph. The Hausdorff distance is a relatively new measure of the similarity of graphs. The notion of the Hausdorff distance considers a special kind of common subgraph of the compared graphs and depends on the structural properties outside of the common subgraph. We study the Hausdorff distance between certain families of graphs that often appear in chemical graph theory. Next to a few results for general graphs, we determine formulae for the distance between paths and cycles. Previously, there was no known efficient algorithm for the problem of determining the Hausdorff distance between two trees, and in this dissertation we present a polynomial-time algorithm for it. The algorithm is recursive and it utilizes the divide and conquer technique. As a subtask it also uses a procedure that is based on the well-known graph algorithm for finding a maximum bipartite matching. The edge metric dimension is a graph invariant that deals with distinguishing the edges of a graph. Let $G=(V(G),E(G))$ be a connected graph, let $w \in V(G)$ be a vertex, and let $e=uv \in E(G)$ be an edge. The distance between the vertex $w$ and the edge $e$ is given by $d_G(e,w)=\min\{d_G(u,w),d_G(v,w)\}$. A vertex $w \in V(G)$ distinguishes two edges $e_1,e_2 \in E(G)$ if $d_G(w,e_1) \ne d_G(w,e_2)$. A set $S$ of vertices in a connected graph $G$ is an edge metric generator of $G$ if every two distinct edges of $G$ are distinguished by some vertex of $S$. The smallest cardinality of an edge metric generator of $G$ is called the edge metric dimension and is denoted by $dim_e(G)$. The concept of the edge metric dimension is new. We study its mathematical properties. We make a comparison between the edge metric dimension and the standard metric dimension of graphs while presenting some realization results concerning the two. We prove that computing the edge metric dimension of connected graphs is NP-hard and give some approximation results. Moreover, we present bounds and closed formulae for the edge metric dimension of several classes of graphs. The mixed metric dimension is a graph invariant similar to the edge metric dimension that deals with distinguishing the elements (vertices and edges) of a graph. A vertex $w \in V(G)$ distinguishes two elements of a graph $x,y \in E(G)\cup V(G)$ if $d_G(w,x) \ne d_G(w,y)$. A set $S$ of vertices in a connected graph $G$ is a mixed metric generator of $G$ if every two elements $x,y \in E(G) \cup V(G)$ of $G$, where $x \neq y$, are distinguished by some vertex of $S$. The smallest cardinality of a mixed metric generator of $G$ is called the mixed metric dimension and is denoted by $dim_m(G)$. In this dissertation, we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results on the mixed metric dimension of certain families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.
Ključne besede: Hausdorff distance, distance between graphs, graph algorithms, trees, graph similarity, edge metric dimension, edge metric generator, mixed metric dimension, metric dimension
Objavljeno v DKUM: 03.08.2020; Ogledov: 1465; Prenosov: 115
.pdf Celotno besedilo (800,48 KB)

5.
Prediction of the hardness of hardened specimens with a neural network
Matej Babič, Peter Kokol, Igor Belič, Peter Panjan, Miha Kovačič, Jože Balič, Timotej Verbovšek, 2014, izvirni znanstveni članek

Opis: In this article we describe the methods of intelligent systems to predict the hardness of hardened specimens. We use the mathematical method of fractal geometry in laser techniques. To optimize the structure and properties of tool steel, it is necessary to take into account the effect of the self-organization of a dissipative structure with fractal properties at a load. Fractal material science researches the relation between the parameters of fractal structures and the dissipative properties of tool steel. This paper describes an application of the fractal dimension in the robot laser hardening of specimens. By using fractal dimensions, the changes in the structure can be determined because the fractal dimension is an indicator of the complexity of the sample forms. The tool steel was hardened with different speeds and at different temperatures. The effect of the parameters of robot cells on the material was better understood by researching the fractal dimensions of the microstructures of hardened specimens. With an intelligent system the productivity of the process of laser hardening was increased because the time of the process was decreased and the topographical property of the material was increased.
Ključne besede: fractal dimension, fractal geometry, neural network, prediction, hardness, steel, tool steel, laser
Objavljeno v DKUM: 17.03.2017; Ogledov: 1860; Prenosov: 110
.pdf Celotno besedilo (632,41 KB)
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6.
Fibonacci dimension of the resonance graphs of catacondensed benzenoid graphs
Aleksander Vesel, 2013, izvirni znanstveni članek

Opis: The Fibonacci dimension ▫$text{fdim}(G)$▫ of a graph ▫$G$▫ was introduced [in S. Cabello, D. Eppstein, S. Klavžar, The Fibonacci dimension of a graph Electron. J. Combin., 18 (2011) P 55, 23 pp] as the smallest integer ▫$d$▫ such that ▫$G$▫ admits an isometric embedding into ▫$Gamma_d$▫, the ▫$d$▫-dimensional Fibonacci cube. The Fibonacci dimension of the resonance graphs of catacondensed benzenoid systems is studied. This study is inspired by the fact, that the Fibonacci cubes are precisely the resonance graphs of a subclass of the catacondensed benzenoid systems. Our results show that the Fibonacci dimension of the resonance graph of a catacondensed benzenoid system ▫$G$▫ depends on the inner dual of ▫$G$▫. Moreover, we show that computing the Fibonacci dimension can be done in linear time for a graph of this class.
Ključne besede: Fibonaccijeva dimenzija, benzenoidni sistemi, resonančni grafi, algoritem, Fibonacci dimension, benzenoid systems, resonance graphs, algorithm
Objavljeno v DKUM: 10.07.2015; Ogledov: 1095; Prenosov: 84
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7.
On the Fibonacci dimension of partial cubes
Aleksander Vesel, 2009

Opis: The Fibonacci dimension fdim▫$(G)$▫ of a graph ▫$G$▫ was introduced in [S. Cabello, D. Eppstein and S. Klavžar, The Fibonacci dimension of a graph, submitted] as the smallest integer ▫$d$▫ such that $G$ admits an isometric embedding into ▫$Q_d$▫, the ▫$d$▫-dimensional Fibonacci cube. A somewhat new combinatorial characterization of the Fibonacci dimension is given, which enables more comfortable proofs of some previously known results. In the second part of the paper the Fibonacci dimension of the resonance graphs of catacondensed benzenoid systems is studied. This study is inspired by the fact, that the Fibonacci cubes are precisely the resonance graphs of a subclass of the catacondensed benzenoid systems. The main result shows that the Fibonacci dimension of the resonance graph of a catacondensed benzenoid system ▫$G$▫ depends on the inner dual of ▫$G$▫. Moreover, we show that computing the Fibonacci dimension can be done in linear time for a graph of this class.
Ključne besede: matematika, teorija grafov, Fibonaccijeva dimenzija, delne kocke, resonančni grafi, benzenoidni sistemi, mathematics, graph theory, Fibonacci dimension, partial cubes, resonance graphs, benzenoid systems
Objavljeno v DKUM: 10.07.2015; Ogledov: 1116; Prenosov: 37
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8.
9.
On induced and isometric embeddings of graphs into the strong product of paths
Janja Jerebic, Sandi Klavžar, 2006, izvirni znanstveni članek

Opis: Primerjani sta krepka izometrična dimenzija in sosedna izometrična dimenzija grafov. Koncepta sta ekvivalentna za grafe premera 2 in v tem primeru se problem določitve dimenzije reducira na problem pokritja s polnimi dvodelnimi grafi. S pomočjo tega pristopa je določena krepka izometrična dimenzija in sosedna izometrična dimenzija za različne grafe (na primer za Petersenov graf). Podan je pozitiven odgovor na Problem 4.1 iz [Fitzpatrick, Nowakowski, The strong isometric dimension of finite reflexive graphs, Discuss. Math. Graph Theory 20 (2000) 23-38], ali obstaja tak graf ▫$G$▫, ki ima krepko izometrično dimenzijo večjo od ▫$lceil |V(G)|/2 rceil$▫.
Ključne besede: matematika, teorija grafov, krepki produkt grafov, krepka izometrična dimenzija, sosedna izometrična dimenzija, mathematics, graph theory, strong product of graphs, adjacent isometric dimension, strong isometric dimension
Objavljeno v DKUM: 10.07.2015; Ogledov: 1137; Prenosov: 104
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10.
On a connection between the VAK, knot theory and El Naschie's theory of the mass spectrum of the high energy elementary particles
Leila Marek-Crnjac, 2004, izvirni znanstveni članek

Opis: Uvedemo teorijo Cantorjevega prostor-časa. V tej teoriji je vsak delec možno interpretirati kot razcep nekega drugega. Nekateri delci so razcepni s protonom in so izraženi s ▫$phioverline{alpha_0}$▫. Če sledimo idejam El Naschieja so limitne množice Kleinove grupe Cantorjeve množice, s Haussdorffovo dimenzijo ▫$phi$▫ ali ▫$frac{1}{phi}, frac{1}{phi^2}, frac{1}{phi^3}...$▫ Z uporabo E-neskončne teorije je masni spekter elementarnih delcev, kot funkcija zlatega reza, v limitni množici Mobius-Kleinove geometrije kvantnega prostor-časa, kot je bilo obravnavano pri Dattu.
Ključne besede: E-neskončna teorija, Hausdorffova dimenzija, Cantorjeva množica, Mobius-Kleinova transformacija, E-infinity theory, Haussdorff dimension, Cantor set, Mobius-Klein transformation
Objavljeno v DKUM: 10.07.2015; Ogledov: 1002; Prenosov: 88
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