Janusz J. Charatonik, Krzysztof Omiljanowski, Biagio Ricceri · SHILAP Revista de lepidopterología · 1992
<p style="font-style: normal;"><span style="font-family: DejaVu Sans,sans-serif;">A pair <em>(X,Y) </em>of topological spaces X and Y<em> </em>is said to have the graph intersection property provided that for each continuous function <em>g: XY</em>, if a connected subset of<em> X Y </em>projects onto the whole <em>Y</em>, then it intersect the graph of <em>g</em>. Various relations between this and other known properties related to mapping theory are studied.</span> <p style="font-style: normal;"><span style="font-family: DejaVu Sans,sans-serif;">In particular, it is proved that: 1) if a space <em>X</em> is completely regular and a space <em>Y </em>is an arcwise connected metric continuum distinct from an arc, then the pair <em>(X,Y)</em> has the graph intersection property if and only if <em>X</em> is hereditary disconnected; 2) if a connected space <em>Y</em> is fixed, then the graph intersection property holds for every pair <em>(X,Y)</em> if and only if there is a closed linear order on <em>Y</em> with minimal and maximal elements. Related results are obtained.</span>
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S. Madhumitha, Sudev Naduvath · Journal of Discrete Mathematical Sciences and Cryptography · 2025
Studies in algebraic graph theory showcase the interplay between group theory and graph theory by defining graphs on groups, investigating their properties, and also by analysing the automorphism groups that emerge from the graphs. In this article, we introduce the idea of constructing an algebraic derived graph; that is, constructing a graph based on the algebraic properties of a graph, by introducing the invariant intersection graph of a graph, constructed based on the automorphism group of a graph. Here, we introduce the graph construction and initiate an investigation on the structure of the invariant intersection graph with respect to the graph and its automorphism group.
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Kriel, C.W. · University of the Witwatersrand, Johannesburg Institutional Repository on DSpace (University of the Witwatersrand, Johannesburg) · 2021
A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2021
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Jeremy Spinrad · American Mathematical Society eBooks · 2003
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Journal of Mathematical and Computational Science · 2020
Let R be a commutative ring with a non-zero identity. In this paper, we define a new graph, the compressed intersection annihilator graph, denoted by IA(R), and investigate some of its properties and its relation with the structure of the ring. It is a generalization of the torsion graph ΓR(R) and has some advantages over the torsion graph and some other graphs. We study classes of rings for which the equivalence between the set of zero-divisors of R, Z(R), being an ideal and the completeness of IA(R) holds. We also study the relation between ΓR(R) and IA(R). Besides, we show that if the compressed intersection annihilator graph of a ring R is finite, then there exists a subring S of R such that IA(S) ∼= IA(R). Also, we show that the compressed intersection annihilator graph will never be a complete bipartite graph. Besides, we show that the graph IA(R) with at least three vertices is connected and its diameter is less than or equal to three. Finally, we determine the properties of the graph in the cases when R is the ring of integers modulo n, the direct product of integral domains, the direct product of Artinian local rings and the direct product of two rings such that one of them is not an integral domain.
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Mindaugas Bloznelis, Michał Karoński · Internet Mathematics · 2014
Vertices of an affiliation network are linked to features and two vertices are declared adjacent whenever they share a common feature. We introduce a random intersection graph process aimed at modeling sparse evolving affiliation networks. We establish the asymptotic degree distribution and provide explicit asymptotic formulas for assortativity and clustering coefficients and show how these edge dependence characteristics vary over time.
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A. Mahmoodi, Alireza Vahidi, Raufeh Manaviyat et al. · Collectanea mathematica · 2023
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Petr Chmel · Digital Repository (National Repository of Grey Literature) · 2022
String graphs are the intersection graphs of curves in the plane. Asinowski et al. [JGAA 2012] introduced a hierarchy of VPG graphs based on the number of bends and showed that the hierarchy contains precisely all string graphs. A similar hierarchy can be observed with k-string graphs: string graphs with the additional condition that each pair of curves has at most k intersection points. We continue in this direction by introducing precisely-k-string graphs which restrict the representation even more so that each pair of curves has either 0 or precisely k intersection points with all of them being crossings. We prove that for each k ≥ 1, any precisely-k-string graph is a precisely-(k + 2)-string graph and that the classes of precisely-k-string graphs and precisely-(k + 1)-string graphs are incomparable with respect to inclusion. We also investigate the problem of finding an efficiently representable class of intersec- tion graphs of objects in the plane that contains all graphs with fixed maximum degree. In the process, we introduce a new hierarchy of intersection graphs of unions of d hori- zontal or vertical line segments, called impure-d-line graphs, and other variations of the class with representation restrictions. We prove that all graphs with maximum degree ≤ 2d are impure-d-line graphs and for d = 1...
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FACHINI, Emanuela, G. Cohen · IRIS Research product catalog (Sapienza University of Rome) · 2009
We introduce a new class of problems lying halfway \nbetween questions about graph capacity and intersection. We say \nthat two binary sequences x and y of the same length have a \nskewincidence if there is a coordinate i for which xi = yi+1 = 1 \nor vice versa. We give relatively close bounds on the maximum \nnumber of binary sequences of length n any pair of which has \na skewincidence. A systematic study of these problems helps \nto understand the mathematical difficulties to solve zero-error \nproblems in information theory.
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Guilherme C. M. Gomes, Marina Groshaus, Carlos V. G. C. Lima et al. · Discrete Applied Mathematics · 2020
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Zhiyun Cheng · Proceedings of the American Mathematical Society · 2024
We prove that two virtual knots have equivalent intersection graphs if and only if they have the same writhe polynomial.
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Benjamin Paul Chamberlain (4775052), Josh Levy-Kramer (4775055), Clive Humby (4775046) et al. · Figshare · 2018
<p>Interesting vertices are depicted as larger red nodes, and the neighbors as smaller, more numerous gray nodes. <b>A</b> shows a complete social network. <b>B</b> depicts the overlapping bipartite neighborhood graphs of the three interesting vertices in <b>A</b>. <b>C</b> summarizes the social network in <b>A</b> by an inferred network using the Jaccard similarity measure of the set of neighboring vertices as edge weights. Vertices connected by high weights are more likely to be in the same community.</p>
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Bohdan Zelinka · Časopis pro pěstování matematiky · 1981
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Mehdi S. Abbas, Hiba Ali Salman · International Journal of Contemporary Mathematical Sciences · 2017
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2016
Let V be a left R-module where R is a (not necessarily commutative) ring with unit. The intersection graph [Formula: see text] of proper R-submodules of V is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper R-submodules of V, and there is an edge between two distinct vertices U and W if and only if U ∩ W ≠ 0. We study these graphs to relate the combinatorial properties of [Formula: see text] to the algebraic properties of the R-module V. We study connectedness, domination, finiteness, coloring, and planarity for [Formula: see text]. For instance, we find the domination number of [Formula: see text]. We also find the chromatic number of [Formula: see text] in some cases. Furthermore, we study cycles in [Formula: see text], and complete subgraphs in [Formula: see text] determining the structure of V for which [Formula: see text] is planar.
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N. R. Aravind, C. R. Subramanian · Discussiones Mathematicae Graph Theory · 2018
We show that the intersection dimension of graphs with respect to several hereditary properties can be bounded as a function of the maximum degree. As an interesting special case, we show that the circular dimension of a graph with maximum degree is at most O log log log . It is also shown that permutation dimension of any graph is at most (log ) 1+o We also obtain bounds on intersection dimension in terms of treewidth.
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Takao Fujita · ChemRxiv · 2025
Graph theory studies mathematical structures composed of vertices and edges in order to model relationships and connectivity [1, 2]. A MetaGraph is a higher-level construct in which the vertices are themselves graphs, and the edges represent specified relations among those graphs. An Iterated MetaGraph extends this concept recursively: its vertices are MetaGraphs, producing a hierarchy of graph-of-graphs structures across multiple levels. In this work, we explore the possibility of extending Mixed Graphs, DiHyperGraphs, Knowledge Graphs, Intersection Graphs, and Chemical Graphs within the framework of MetaGraphs and Iterated MetaGraphs.
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Jhon Cris Bonifacio, Clarence Joy Andaya, Daryl M. Magpantay · European Journal of Pure and Applied Mathematics · 2023
This paper defines a new class of graphs using the spanning subgraphs of a cycle graph as vertices. This class of graphs is called $j$-edge intersection graph of cycle graph, denoted by $E_{C_{(n,j)}}$. The vertex set of $E_{C_{(n,j)}}$ is the set of spanning subgraphs of cycle graph with $j$ edges where $n \geq 3$ and $j$ is a nonnegative integer such that $1 \leq j \leq n$. Moreover, two distinct vertices are adjacent if they have exactly one edge in common. $E_{C_{(n,j)}}$ is considered as a simple graph. Furthermore, $E_{C_{(n,j)}}$ is characterized by the value of $j$ that is when $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$ and $2 \leq j \leq \lceil \frac{n}{2} \rceil$. When $j=1$ or $\lceil \frac{n}{2} \rceil < j \leq n$, the new graph only produced an empty graph. Hence, the proponents only considered the value when $2 \leq j \leq \lceil \frac{n}{2} \rceil$ in determining the order and size of $E_{C_{(n,j)}}$. Moreover, this paper discusses necessary and sufficient conditions where the $j$-edge intersection graph of $C_n$ is isomorphic to the cycle graph. Furthermore, the researchers determined a lower bound for the independence number, and an upper bound for the domination number of $E_{C_{(n,j)}}$ when $j=2$.
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T. Asir, T. Tamizh Chelvam · Discussiones Mathematicae Graph Theory · 2012
In this paper, we consider the intersection graph I (Z n ) of gamma sets in the total graph on Z n . We characterize the values of n for which I (Z n ) is complete, bipartite, cycle, chordal and planar. Further, we prove that I (Z n ) is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of I (Z n ).
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Discussiones Mathematicae, T. Tamizh Chelvam, T. Asir · 2016
In this paper, we consider the intersection graph IΓ(Zn) of gamma sets in the total graph on Zn. We characterize the values of n for which IΓ(Zn) is complete, bipartite, cycle, chordal and planar. Further, we prove that IΓ(Zn) is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of IΓ(Zn).
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