Complete Graph Research Articles

Counterexamples to Gerbner's conjecture on stability of maximal F‐free graphs

AbstractLet and let be an ‐chromatic graph with a color‐critical edge, that is, there exists such that has chromatic number . Gerbner recently conjectured that every ‐vertex maximal ‐free graph with at least edges contains an induced complete ‐partite graph on vertices. Let be a graph obtained from copies of by sharing a common edge. In this paper, we show that for all if is an ‐vertex maximal ‐free graph with at least edges, then contains an induced complete bipartite graph on vertices. We also construct graphs to show that the magnitude in is best possible, which disproves Gerbner's conjecture for .

Wiener index of an ideal-based zero-divisor graph of commutative ring with unity

The Wiener index of a connected graph G is W ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u , v ) . In this paper, we obtain the Wiener index of H-generalized join of graphs G 1 , G 2 , … , G k . As a consequence, we obtain some earlier known results in [Alaeiyan et al. in Aust. J. Basic Appl. Sci. (2011) 5(12): 145–152; Yeh et al. in Discrete Math. (1994) 135: 359–365] and we also obtain the Wiener index of the generalized corona product of graphs. We further show that the ideal-based zero-divisor graph Γ I ( R ) is a H-generalized join of complete graphs and totally disconnected graphs. As a result, we find the Wiener index of the ideal-based zero-divisor graph Γ I ( R ) and we deduce some of the main results in [Selvakumar et al. in Discrete Appl. Math. (2022) 311: 72–84]. Moreover, we show that W ( Γ I ( Z n ) ) is a quadratic polynomial in n, where Z n is the ring of integers modulo n and we calculate the exact value of the Wiener index of Γ Nil ( R ) ( R ) , where Nil(R) is nilradical of R. Furthermore, we give a Python program for computing the Wiener index of Γ I ( Z n ) if I is an ideal of Z n generated by pr , where pr is a proper divisor of n, p is a prime number and r is a positive integer with r ≥ 2 .

Total k-domination in Cartesian product of complete graphs

Let G=(V,E) be a finite undirected graph. A set S of vertices in V is said to be total k-dominating if every vertex in V is adjacent to at least k vertices in S. The total k-domination number, γkt(G), is the minimum cardinality of a total k-dominating set in G. In this work we study the total k-domination number of Cartesian product of two complete graphs which is a lower bound of the total k-domination number of Cartesian product of two graphs. We obtain new lower and upper bounds for the total k-domination number of Cartesian product of two complete graphs. Some asymptotic behaviors are obtained as a consequence of the bounds we found. In particular, lim infn→∞γkt(G□H)n:G,H are graphs of ordern≤2k2−1+k+42−1−1. We also prove that the equality is attained if k is even. The equality holds when G,H are both isomorphic to the complete graph, Kn, with n vertices. Furthermore, we obtain closed formulas for the total 2-domination number of Cartesian product of two complete graphs of whatever order. Besides, we prove that, for k=3, the inequality above is improvable to lim infn→∞γ3t(Kn□Kn)/n≤11/5.

C2IMUFS: Complementary and Consensus Learning-Based Incomplete Multi-View Unsupervised Feature Selection

Multi-view unsupervised feature selection (MUFS) has been demonstrated as an effective technique to reduce the dimensionality of multi-view unlabeled data. The existing methods assume that all of views are complete. However, multi-view data are usually incomplete, i.e., a part of instances are presented on some views but not all views. Besides, learning the complete similarity graph, as an important promising technology in existing MUFS methods, cannot achieve due to the missing views. In this paper, we propose a complementary and consensus learning-based incomplete multi-view unsupervised feature selection method (C <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula> IMUFS) to address the aforementioned issues. Concretely, C <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula> IMUFS integrates feature selection into an extended weighted non-negative matrix factorization model equipped with adaptive learning of view-weights and a sparse <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2,p}$</tex-math></inline-formula> -norm, which can offer better adaptability and flexibility. By the sparse linear combinations of multiple similarity matrices derived from different views, a complementary learning-guided similarity matrix reconstruction model is presented to obtain the complete similarity graph in each view. Furthermore, C <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula> IMUFS learns a consensus clustering indicator matrix across different views and embeds it into a spectral graph term to preserve the local geometric structure. Comprehensive experimental results on real-world datasets demonstrate the effectiveness of C <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula> IMUFS compared with state-of-the-art methods.

Complete characterizations of the 2-domination and P3-hull number polytopes

Given a graph G=(V,E), a subset S⊆V is 2-dominating if every vertex in S¯ has at least two neighbors in S. The minimum cardinality of such a set is called the 2-domination number of G. Consider a process in discrete time that, starting with an initial set of marked vertices S, at each step marks all unmarked vertices having two marked neighbors. In such a process, the minimum number of initial vertices in S such that eventually all vertices are marked is called the P3-hull number of G. In this work, we explore a polyhedral relation between these two parameters and, in addition, we provide new families of valid inequalities for the associated polytopes. Finally, we give explicit descriptions of the polytopes associated to these problems when G is a path, a cycle, a complete graph, or a tree.

Decision Making Under Pythagorean Fuzzy Soft Environment

This research article illustrates the notion of strong and complete Pythagorean fuzzy soft graphs (PFSGs). Different operations on PFSGs including union of two PFSGs, join of two PFSGs, lexicographic product of two PFSGs, strong product of two PFSGs, Cartesian product of two PFSGs, composition of two PFSGs are also analysed here. Some properties related to these products are discussed here. The idea of complememt of a PFSG is also eloborated here. Moreover, we establish the application of PFSG in the decision making (DM) problem.

An Optimal Algorithm for Finding Champions in Tournament Graphs

A tournament graph is a complete directed graph, which can be used to model a round-robin tournament between $n$ players. In this paper, we address the problem of finding a champion of the tournament, also known as Copeland winner, which is a player that wins the highest number of matches. In detail, we aim to investigate algorithms that find the champion by playing a low number of matches. Solving this problem allows us to speed up several Information Retrieval and Recommender System applications, including question answering, conversational search, etc. Indeed, these applications often search for the champion inducing a round-robin tournament among the players by employing a machine learning model to estimate who wins each pairwise comparison. Our contribution, thus, allows finding the champion by performing a low number of model inferences. We prove that any deterministic or randomized algorithm finding a champion with constant success probability requires $\Omega(\ell n)$ comparisons, where $\ell$ is the number of matches lost by the champion. We then present an asymptotically-optimal deterministic algorithm matching this lower bound without knowing $\ell$, and we extend our analysis to three variants of the problem. Lastly, we conduct a comprehensive experimental assessment of the proposed algorithms on a question answering task on public data. Results show that our proposed algorithms speed up the retrieval of the champion up to $13\times$ with respect to the state-of-the-art algorithm that perform the full tournament.

10 problems for partitions of triangle-free graphs

We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erdős’ Sparse Half Conjecture.Among others we prove the following variant of it: For every sufficiently large even integer n the following holds. Every triangle-free graph on n vertices has a partition V(G)=A∪B with |A|=|B|=n/2 such that e(G[A])+e(G[B])≤n2/16. This result is sharp since the complete bipartite graph with class sizes 3n/4 and n/4 achieves equality, when n is a multiple of 4.Additionally, we discuss similar problems for K4-free graphs.

Grundy number of corona product of some graphs

The Grundy number of a graph G is the maximum number k of colors used to color the vertices of G such that the coloring is proper and every vertex u colored with color i, 1 ≤ i ≤ k, is adjacent to i – 1 vertices colored with each color j, 1 ≤ j ≤ i − 1. In this paper we obtain the Grundy number of corona product of some graphs, denoted by G ◦ H. First, we consider the graph G be 2-regular graph and H be a cycle, complete bipartite, ladder graph and fan graph. Also we consider the graph G and H be a complete bipartite graphs, fan graphs.

Partitioning a 2-edge-coloured graph of minimum degree [formula omitted] into three monochromatic cycles

Lehel conjectured in the 1970s that the vertices of every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomassé. However, the host graph G does not have to be complete. It suffices to require that G has minimum degree at least 3n/4, where n is the order of G, as was shown recently by Letzter, confirming a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy. This degree condition is tight.Here we continue this line of research, by proving that for every red and blue edge-colouring of an n-vertex graph of minimum degree at least 2n/3+o(n), there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.

Solving the Shortest Total Path Length Spanning Tree Problem Using the Modified Sollin and Modified Dijkstra Algorithms

In a weighted connected graph, the shortest total path length spanning tree problem is a problem when we need to discover the spanning tree with the lowest total cost of all pairwise distances between its vertices. This problem is also known as the minimum routing cost spanning tree (MRCST). In this study, we will discuss the Modified Sollin and Modified Dijkstra Algorithms to solve that problem which implemented on 300 problems are complete graphs of orders 10 to 100 in increments of 10, where every order consists of 30 problems. The results show that the performance of the Modified Dijkstra and the Modified Sollin Algorithms are slightly similar. On orders 10, 20, 30, 60, and 80, the Modified Dijkstra Algorithm performs better than the Modified Sollin, however on orders 40, 50, 70, 90, and 100, the Modified Sollin performs better.

ANNIHILATING IDEAL AND EXACT ANNIHILATING IDEAL GRAPH OF RING Z_n

The existence of annihilator in the ring motivates the emergence of studies on Annihilating Ideal and Exact Annihilating Ideal Graphs. The purpose of this research is to describe the characteristics of an (exact) annihilating ideal of ring . The method used in this research is literature study. The results of this study discuss finiteness, adjacency, connectedness, vertices, and types of and . Furthermore, the number of vertices of an Annihilating Ideal Graph is determined by the factorization of . The adjacency of two vertices is determined by the divisibleness of . The results also show that is a subgraph of . can be represented as a union of several complete graphs.

Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias.

We study the Galam majority rule dynamics with contrarian behavior and an oscillating external propaganda in a population of agents that can adopt one of two possible opinions. In an iteration step, a random agent interacts with three other random agents and takes the majority opinion among the agents with probability p(t) (majority behavior) or the opposite opinion with probability 1-p(t) (contrarian behavior). The probability of following the majority rule p(t) varies with the temperature T and is coupled to a time-dependent oscillating field that mimics a mass media propaganda, in a way that agents are more likely to adopt the majority opinion when it is aligned with the sign of the field. We investigate the dynamics of this model on a complete graph and find various regimes as T is varied. A transition temperature Tc separates a bimodal oscillatory regime for T<Tc, where the population's mean opinion m oscillates around a positive or a negative value from a unimodal oscillatory regime for T>Tc in which m oscillates around zero. These regimes are characterized by the distribution of residence times that exhibit a unique peak for a resonance temperature T*, where the response of the system is maximum. An insight into these results is given by a mean-field approach, which also shows that T* and Tc are closely related.

Критические показатели спиновой модели на полносвязном графе при наличии антиферромагнитного взаимодействия

рассмотрена спиновая система на полносвязном графе, состоящая из двух взаимодействующих подансамблей: cпины, принадлежащие одну и тому же подансамблю, взаимодействуют ферромагнитным образом, а перекрестное взаимодействие между спинами разных подансамблей – антиферромагнитное. Введено условие сбалансированности системы, означающее, что число ближайших соседей для спина первого подансабля равно числу ближайших соседей у спина во втором подансамбле. Показано, что критические показатели и функция скейлинга сбалансированной системы кардинально отличаются от классических, присущих несбалансированной системе. Полученные результаты подтверждаются Монте-Карло симуляцией трехмерной слоистой спиновой модели, проведенной для сбалансированной и несбалансированной систем. we studied the spin system on the complete graph consisting of two interacting subensembles. The spins that belong to the same ensemble have ferromagnetic interactions; the inter-ensemble interactions are antiferromagnetic. We introduced the “balanced system” term, which defines the equality of the number of the nearest neighbors for spins of different sub-ensembles. It is shown that the critical variables and the scaling function are different from the classical ones of the unbalanced system. These results are confirmed by the Monte-Carlo simulation of the 3D layered spin model for both balanced and unbalanced systems.

Evolution of quantum resources in quantum-walk-based search algorithm

Quantum walk is fundamental to designing many quantum algorithms. Here we consider the effects of quantum coherence and quantum entanglement for the quantum walk search on the complete bipartite graph. First, we numerically show the complementary relationship between the success probability and the two quantum resources (quantum coherence and quantum entanglement). We also provide theoretical analysis in the asymptotic scenarios. At last, we discuss the role played by generalized depolarizing noises and find that it would influence the dynamics of success probability and quantum coherence sharply, which is demonstrated by theoretical derivation and numerical simulation.

Energy of graphs with no eigenvalue in the interval (−1,1)

The energy of a simple graph G of order n, denoted by E(G), is E(G)=|λ1|+⋯+|λn|, where λ1,…,λn are the eigenvalues of G. Recently, it has been conjectured that if all eigenvalues of G are non-zero, then E(G)≥Δ(G)+δ(G) and the equality holds if and only if G is a complete graph, where δ(G) and Δ(G) are the minimum degree and the maximum degree of vertices of G, respectively. Motivated by this conjecture we find some results for energy of graphs in terms of the number of vertices, the number of edges and the nullity. We prove that if G is a graph with n vertices and m edges such that G has no eigenvalue in the interval (−1,1), then E(G)≥n−1+2m−n+1 and the equality holds if and only if G is a disjoint union of some special complete graphs. In particular, we prove that the above conjecture is true for graphs that have no eigenvalue in the interval (−1,1).

A probabilistic model for networks generated by actors’ characteristics

This work presents the affinity network model for random graphs, consisting of a broad family of random graph models depending on some parameters. In this model, we suppose that each individual randomly chooses a set of characteristics that represent him according to a certain probability measure. The connections between two individuals depend on their shared characteristics and are valued according to a function that measures what we call affinity in the network. According to the choice of this function, the network’s density can vary from sparse to complete graph, causing the model to be very flexible, which makes it suitable to fit with real networks. To illustrate the behavior of the affinity network model, we present a Monte Carlo simulation study. We tune the model’s generating parameters, analyze its topological measurements, and compare them with equivalent graphs with edges occurring independently, disregarding actors’ characteristics.

The generalized 4-connectivity of complete-transposition graphs

The fault tolerability of the network is usually measured by the classical or generalized connectivity of the graph. For any subset with , a tree T is called an S-tree if . Furthermore, any two S-tree and are internally disjoint if and . We denote by the maximum number of pairwise internally disjoint S-trees in G. For an integer , the generalized k-connectivity of a graph G is defined as and . In this paper, we establish the generalized 4-connectivity of the Cayley graph generated by complete graphs.

Bounds on the parameters of non- L -borderenergetic graphs

UDC 519.17 We consider graphs such that their Laplacian energy is equivalent to the Laplacian energy of the complete graph of the same order, which is called an L -borderenergetic graph. Firstly, we study the graphs with degree sequence consisting of at most three distinct integers and give new bounds for the number of vertices of these graphs to be non- L -borderenergetic. Second, by using Koolen–Moulton and McClelland inequalities, we give new bounds for the number of edges of a non- L -borderenergetic graph. Third, we use recent bounds given by Milovanovic, et al. on Laplacian energy to get similar conditions for non- L -borderenergetic graphs. Our bounds depend only on the degree sequence of a graph, which is much easier than computing the spectrum of the graph. In other words, we developed a faster approach to exclude non- L -borderenergetic graphs.

Generalizations of some Nordhaus–Gaddum‐type results on spectral radius

AbstractLet be a simple graph and the spectral radius of . For , a ‐edge decomposition is spanning subgraphs such that their edge sets form a ‐partition of the edge set of . In this paper, we obtain some sharp lower and upper bounds for in terms of the clique number of and the size of , and discuss what ‐edge decomposition can maximize when is a complete graph. These generalize some Nordhaus–Gaddum‐type results on spectral radius for , due to Nosal, Hong and Shu, and Nikiforov.