Complete Graph Research Articles

A multipopulation multitasking evolutionary scheme with adaptive knowledge transfer to solve the clustered minimum routing cost tree problem

The advancement in communication technology nowadays has increased the number of networked devices exponentially. Thus, network applications begin to cluster terminals to address scalability, privacy, and device connectivity with low routing costs. A recently introduced NP-hard problem called the Clustered Minimum Routing Cost Tree (CluMRCT) problem is assessed to be a fundamental key in designing an efficient network with low routing costs. Lately, some evolutionary multitasking meta-heuristics have been proposed to tackle multiple CluMRCT problems concurrently. However, these previous studies still have some limitations, such as only being applicable to complete graphs or there is no mechanism for effective knowledge transfer between problems when solving them simultaneously. To overcome these limitations, this paper proposes a novel multi-population multitasking evolutionary framework to address the problem. The individual representation is developed based on Network random key scheme and can be applied to both sparse and complete graphs. Moreover, our proposal can adaptively exploit positive knowledge transfer by adjusting the number of individuals migrating from other populations to a particular task. Then, the proposed algorithm is experimented on various data types and compared with other state-of-the-art algorithms. The experimental results show the proposal's effectiveness in most cases of the dataset, followed by detailed comparison, evaluation, and analysis.

On minimum spanning trees for random Euclidean bipartite graphs

AbstractWe consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$ -th power of their Euclidean distance, with $p\gt 0$ . In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$ , we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$ , we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

Graph-Based Generalization of Galam Model: Convergence Time and Influential Nodes

We study a graph-based generalization of the Galam opinion formation model. Consider a simple connected graph which represents a social network. Each node in the graph is colored either blue or white, which indicates a positive or negative opinion on a new product or a topic. In each discrete-time round, all nodes are assigned randomly to groups of different sizes, where the node(s) in each group form a clique in the underlying graph. All the nodes simultaneously update their color to the majority color in their group. If there is a tie, each node in the group chooses one of the two colors uniformly at random. Investigating the convergence time of the model, our experiments show that the convergence time is a logarithm function of the number of nodes for a complete graph and a quadratic function for a cycle graph. We also study the various strategies for selecting a set of seed nodes to maximize the final cascade of one of the two colors, motivated by viral marketing. We consider the algorithms where the seed nodes are selected based on the graph structure (nodes’ centrality measures such as degree, betweenness, and closeness) and the individual’s characteristics (activeness and stubbornness). We provide a comparison of such strategies by conducting experiments on different real-world and synthetic networks.

Occupancy probabilities in superintegrable bosonic networks

We associate a bosonic network to each complete bipartite graph. A Hamiltonian is defined with hopping terms on the edges of the graph, and a global interaction term depending on the vertex sets. For a generic graph this Hamiltonian is superintegrable, and we derive a Bethe Ansatz solution for the energy and eigenstates. These results provide the means to investigate the quantum dynamics and allow for the computation of occupancy probabilities in certain regimes. We use these results to gain an understanding of entanglement evolution in the network.

More Ramsey theory for highly connected monochromatic subgraphs

Abstract An infinite graph is said to be highly connected if the induced subgraph on the complement of any set of vertices of smaller size is connected. We continue the study of weaker versions of Ramsey’s theorem on uncountable cardinals asserting that if we color edges of the complete graph, we can find a large highly connected monochromatic subgraph. In particular, several questions of Bergfalk, Hrušák, and Shelah (2021, Acta Mathematica Hungarica 163, 309–322) are answered by showing that assuming the consistency of suitable large cardinals, the following are relatively consistent with ZFC: • $\kappa \to _{hc} (\kappa )^2_\omega $ for every regular cardinal $\kappa \geq \aleph _2$ , • $\neg \mathsf {CH}+ \aleph _2 \to _{hc} (\aleph _1)^2_\omega $ . Building on a work of Lambie-Hanson (2023, Fundamenta Mathematicae. 260(2):181–197), we also show that • $\aleph _2 \to _{hc} [\aleph _2]^2_{\omega ,2}$ is consistent with $\neg \mathsf {CH}$ . To prove these results, we use the existence of ideals with strong combinatorial properties after collapsing suitable large cardinals.

Statistics of Feynman amplitudes in ϕ4-theory

The amplitude of subdivergence-free logarithmically divergent Feynman graphs in ϕ4-theory in 4 spacetime dimensions is given by a single number, the Feynman period. We numerically compute the periods of 1.3 million completed graphs, this represents more than 33 million graphs contributing to the beta function. Our data set includes all primitive graphs up to 13 loops, and non-complete samples up to 18 loops, with an accuracy of ca. 4 significant digits.We implement all known symmetries of the period in a new computer program and count them up to 14 loops. Combining the symmetries, we discover relations between periods that had been overlooked earlier. All expected symmetries are respected by the numerical values of periods.We examine the distribution of the numerically computed Feynman periods. We confirm the leading asymptotic growth of the average period with growing loop order, up to a factor of 2. At high loop order, a limiting distribution is reached for the amplitudes near the mean. A small class of graphs, most notably the zigzags, grows significantly faster than the mean and causes the limiting distribution to have divergent moments even when normalized to unit mean. We examine the relation between the period and various properties of the underlying graphs. We confirm the strong correlation with the Hepp bound, the Martin invariant, and the number of 6-edge cuts. We find that, on average, the amplitude of planar graphs is significantly larger than that of non-planar graphs, irrespective of O(N) symmetry.We estimate the primitive contribution to the 18-loop beta function of the O(N)-symmetric theory. We find that primitive graphs constitute a large part of the beta function in MS for L → ∞ loops. The relative contribution of planar graphs increases with growing N and decreases with growing loop order L.

The saturation number of K3,3

A graph G is called F-saturated if G does not contain F as a subgraph (not necessarily induced) but the addition of any missing edge to G creates a copy of F. The saturation number of F, denoted by sat(n,F), is the minimum number of edges in an n-vertex F-saturated graph. Determining the saturation number of complete bipartite graphs is one of the most important problems in the study of saturation numbers. The value of sat(n,K2,2) was shown to be ⌊3n−52⌋ by Ollmann, and a shorter proof was later given by Tuza. For K2,3, there has been a series of study aiming to determine sat(n,K2,3) over the years. This was finally achieved by Chen who confirmed a conjecture of Bohman, Fonoberova, and Pikhurko that sat(n,K2,3)=2n−3 for all n≥5. Pikhurko and Schmitt conjectured that sat(n,K3,3)=(3+o(1))n. In this paper, for n≥9, we give an upper bound of 3n−9 for sat(n,K3,3), and prove that 3n−9 is also a lower bound when the minimum degree of a K3,3-saturated graph is 2 or 5, where it is trivial when the minimum degree is greater than 5.

Induced subgraphs and tree decompositions V. one neighbor in a hole

AbstractWhat are the unavoidable induced subgraphs of graphs with large treewidth? It is well‐known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the “basic treewidth obstructions”). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so‐called “layered wheels,” contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.

Proper Cycles and Rainbow Cycles in 2-triangle-free edge-colored Complete Graphs

Proper Cycles and Rainbow Cycles in 2-triangle-free edge-colored Complete Graphs

A graph minor condition for graphs to be [formula omitted]-linked

A graph is called k-linked, if for any 2k distinct vertices x1,x2,…,xk,y1,y2,…,yk, there exist k vertex disjoint paths P1,P2,…,Pk such that Pi connects xi and yi for each 1≤i≤k. Robertson and Seymour showed that every 2k-connected graph having K3k as a minor is k-linked. In 2005, Chen, Gould, Kawarabayashi, Pfender, and Wei proved that every 6-connected graph having K9− as a minor is 3-linked, where Kk−i is the graph obtained from the complete graph with k vertices by deleting exactly i edges. We improve these two results by showing that every 2k-connected graph having the graph obtained from K3k by deleting independent k edges as a minor is k-linked.

Coloring ([formula omitted], kite)-free graphs with small cliques

Let Pn and Kn denote the induced path and complete graph on n vertices, respectively. The kite is the graph obtained from a P4 by adding a vertex and making it adjacent to all vertices in the P4 except one vertex with degree 1. A graph is (P5, kite)-free if it has no induced subgraph isomorphic to a P5 or a kite. For a graph G, the chromatic number of G (denoted by χ(G)) is the minimum number of colors needed to color the vertices of G such that no two adjacent vertices receive the same color, and the clique number of G is the size of a largest clique in G. Here, we are interested in coloring the class of (P5, kite)-free graphs with small clique number. It is known that every (P5, kite, K3)-free graph G satisfies χ(G)≤3, every (P5, kite, K4)-free graph G satisfies χ(G)≤4, and that every (P5, kite, K5)-free graph G satisfies χ(G)≤6. In this paper, we show the following: •Every (P5, kite, K6)-free graph G satisfies χ(G)≤7.•Every (P5, kite, K7)-free graph G satisfies χ(G)≤9. We also give examples to show that the above bounds are tight. Based on these partial results and some other examples, we conjecture that every (P5, kite)-free graph G satisfies χ(G)≤⌊3ω(G)2⌋, and that the bound is tight.

Finding Large Rainbow Trees in Colourings of $K_{n,n}$

A subgraph of an edge-coloured graph is called rainbow if all of its edges have distinct colours. An edge-colouring is called locally $k$-bounded if each vertex is incident with at most $k$ edges of the same colour. Recently, Montgomery, Pokrovskiy and Sudakov showed that for large $n$, a certain locally 2-bounded edge-colouring of the complete graph $K_{2n+1}$ contains a rainbow copy of any tree with $n$ edges, thereby resolving a long-standing conjecture by Ringel: For large $n$, $K_{2n+1}$ can be decomposed into copies of any tree with $n$ edges. In this paper, we employ their methods to show that any locally $k$-bounded edge-colouring of the complete bipartite graph $K_{n,n}$ contains a rainbow copy of any tree $T$ with $(1- o(1))n/k$ edges. We show that this implies that every tree with $n$ edges packs at least $n$ times into $K_{n+o(1),n+o(1)}$. We conjecture that for large $n$, $K_{n,n}$ can be decomposed into $n$ copies of any tree with $n$ edges.

The rotation distance of brooms

The associahedron A(G) of a graph G has the property that its vertices can be thought of as the search trees on G and its edges as the rotations between two search trees. If G is a simple path, then A(G) is the usual associahedron and the search trees on G are binary search trees. Computing distances in the graph of A(G), or equivalently, the rotation distance between two binary search trees, is a major open problem. Here, we consider the different case when G is a complete split graph. In that case, A(G) interpolates between the stellohedron and the permutohedron, and all the search trees on G are brooms. We show that the rotation distance between any two such brooms and therefore the distance between any two vertices in the graph of the associahedron of G can be computed in quasi-quadratic time in the number of vertices of G.

The spectral radius of minor-free graphs

In this paper, we present a sharp upper bound for the spectral radius of an n-vertex F-minor-free graph for sufficiently large n, where F is obtained from the complete graph Kr by deleting disjoint paths. Furthermore, the graphs which achieve the sharp bound are characterized. This result may be regarded as a spectral extremal analogue of the number of edges in an n-vertex F-minor-free graph.

A Hamilton Path-Based Approach to DNA Sequence Analysis

The Hamiltonian path is a famous algorithm for determining the existence or nonexistence of a graph. However, for certain types of graphs, we can apply certain conditions to determine its existence. In this paper, we discuss the results show that there can be at most (n-1)/2 edges disjoint Hamiltonian paths in a complete graph. Also, discussed examples of its applications in different fields. Furthermore, the application of Hamilton path in the DNA sequence has been investigated.

A continuous-time N-interaction random graph model

In this paper a continuous-time evolving random graph model is defined and examined. The main units of the model are complete graphs on N vertices, where N ≥ 3 is a fixed integer. At each birth event a new vertex and random number of edges are added to the graph. The asymptotic behaviour of the number of vertices and the asymptotic behaviour of the number of m-cliques ( 2 ≤ m ≤ N ) are studied. The proofs are based on general results of the theory of branching processes.

Mapping Connectivity Patterns: Degree-Based Topological Indices of Corona Product Graphs

Graph theory (GT) is a mathematical field that involves the study of graphs or diagrams that contain points and lines to represent the representation of mathematical truth in a diagrammatic format. From simple graphs, complex network architectures can be built using graph operations. Topological indices (TI) are graph invariants that correlate the physicochemical and interesting properties of different graphs. TI deal with many properties of molecular structure as well. It is important to compute the TI of complex structures. The corona product (CP) of two graphs G and H gives us a new graph obtained by taking one copy of G and V G copies of H and joining the i th vertex of G to every vertex in the i th copy of H . In this paper, based on various CP graphs composed of paths, cycles, and complete graphs, the geometric index (GA) and atom bond connectivity (ABC) index are investigated. Particularly, we discussed the corona products P s ⨀ P t , C t ⨀ C s , K t ⊙ K s , K t ⊙ P s , and P s ⊙ K t and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical findings supported by proofs.

Logical–Mathematical Foundations of a Graph Query Framework for Relational Learning

Relational learning has attracted much attention from the machine learning community in recent years, and many real-world applications have been successfully formulated as relational learning problems. In recent years, several relational learning algorithms have been introduced that follow a pattern-based approach. However, this type of learning model suffers from two fundamental problems: the computational complexity arising from relational queries and the lack of a robust and general framework to serve as the basis for relational learning methods. In this paper, we propose an efficient graph query framework that allows for cyclic queries in polynomial time and is ready to be used in pattern-based learning methods. This solution uses logical predicates instead of graph isomorphisms for query evaluation, reducing complexity and allowing for query refinement through atomic operations. The main differences between our method and other previous pattern-based graph query approaches are the ability to evaluate arbitrary subgraphs instead of nodes or complete graphs, the fact that it is based on mathematical formalization that allows the study of refinements and their complementarity, and the ability to detect cyclic patterns in polynomial time. Application examples show that the proposed framework allows learning relational classifiers to be efficient in generating data with high expressiveness capacities. Specifically, relational decision trees are learned from sets of tagged subnetworks that provide both classifiers and characteristic patterns for the identified classes.

Monochromatic spanning trees and matchings in ordered complete graphs

AbstractWe study two well‐known Ramsey‐type problems for (vertex‐)ordered complete graphs. Two independent edges in ordered graphs can be nested, crossing, or separated. Apart from two trivial cases, these relations define six types of subgraphs, depending on which one (or two) of these relations are forbidden. Our first target is to refine a remark by Erdős and Rado that every 2‐coloring of the edges of a complete graph contains a monochromatic spanning tree. We show that forbidding one relation we always have a monochromatic (nonnested, noncrossing, and nonseparated) spanning tree in a 2‐edge‐colored ordered complete graph. On the other hand, if two relations are forbidden, then it is possible that we have monochromatic (nested, separated, and crossing) subtrees of size only in a 2‐colored ordered complete graph on vertices. Some of these results relate to drawings of complete graphs. For instance, the existence of a monochromatic nonnested spanning tree in 2‐colorings of ordered complete graphs verifies a more general conjecture for twisted drawings. Our second subject is to refine the Ramsey number of matchings, that is, pairwise independent edges for ordered complete graphs. Cockayne and Lorimer proved that for given positive integers , is the smallest integer with the following property: every ‐coloring of the edges of a complete graph contains a monochromatic matching with edges. We conjecture that this result can be strengthened: ‐colored ordered complete graphs on vertices contain monochromatic nonnested and also nonseparated matchings with edges. We prove this conjecture for some special cases including the following. (i) Every ‐colored ordered complete graph on vertices contains a monochromatic nonnested matching of size two ( case). We prove it by showing that the chromatic number of the subgraph of the Kneser graph induced by the nonnested 2‐matchings in an ordered complete graph on vertices is ‐chromatic. (ii) Every 2‐colored ordered complete graph on vertices contains a monochromatic nonseparated matching of size ( case). This is the hypergraph analog of a result of Kaiser and Stehlík who proved that the Kneser graph induced by the nonseparated 2‐matchings in an ordered complete graph on vertices is ‐chromatic.For nested, separated, and crossing matchings the situation is different. The smallest ensuring a monochromatic matching of size in every ‐coloring is in the first two cases and one less in the third case.

On Group-Vertex-Magic Labeling of Simple Graphs

Let A be an Abelian group with identity 0. The A-vertex-magic labeling of a graph G is a mapping from the set of vertices in G to A-{0} such that the sum of the labels of every open neighborhood vertex of v is equal, for every vertex v in G. In this article, we discuss group-vertex-magic labeling of some simple graphs by using the Abelian group Zk, with natural numbers k1. We investigated some classes of simple graphs are path graphs, complete graphs, cyclic graphs, and star graphs. The method we used in this article is literature study and then developing the properties of vertex-magic labeling of some simple graphs, that are path graphs, complete graphs, cyclic graphs, and star graphs. We obtain that complete graphs, cyclic graphs, and star graphs have Zk-vertex-magic labeling, while path graphs have vertex-magic labeling only for n=2,3.