Clique Number Research Articles

Maximal cliques in scale-free random graphs

Abstract We investigate the number of maximal cliques, that is, cliques that are not contained in any larger clique, in three network models: Erdős–Rényi random graphs, inhomogeneous random graphs (IRGs) (also called Chung–Lu graphs), and geometric inhomogeneous random graphs (GIRGs). For sparse and not-too-dense Erdős–Rényi graphs, we give linear and polynomial upper bounds on the number of maximal cliques. For the dense regime, we give super-polynomial and even exponential lower bounds. Although (G)IRGs are sparse, we give super-polynomial lower bounds for these models. This comes from the fact that these graphs have a power-law degree distribution, which leads to a dense subgraph in which we find many maximal cliques. These lower bounds seem to contradict previous empirical evidence that (G)IRGs have only few maximal cliques. We resolve this contradiction by providing experiments indicating that, even for large networks, the linear lower-order terms dominate, before the super-polynomial asymptotic behavior kicks in only for networks of extreme size.

Spectral Pseudorandomness and the Road to Improved Clique Number Bounds for Paley Graphs

We study subgraphs of Paley graphs of prime order p induced on the sets of vertices extending a given independent set of size a to a larger independent set. Using a sufficient condition due to Kunisky (2023), we show that estimates on a new family of necklace character sums would imply that, as p → ∞ , the empirical spectral distributions of the adjacency matrices of any sequence of such subgraphs have the same weak limit (after rescaling) as those of subgraphs induced on a random set including each vertex independently with probability 2 − a , namely, a Kesten-McKay law with parameter 2 a . We prove the necessary estimates for a = 1, obtaining in the process an alternate proof of a character sum equidistribution result of Xi (2022), and provide numerical evidence for this weak convergence for a ≥ 2 . We also conjecture that the minimum eigenvalue of any such sequence converges (after rescaling) to the left edge of the corresponding Kesten-McKay law, and provide numerical evidence for this convergence. Finally, we show that, once a ≥ 3 , this (conjectural) convergence of the minimum eigenvalue would imply bounds on the clique number of the Paley graph improving on the current state of the art due to Hanson and Petridis (2021), and that this convergence for all a ≥ 1 would imply that the clique number is o ( p ) .

The Multiplicative Sum Zagreb Indices of Graphs with Given Clique Number

The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph G is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.

Line game-perfect graphs

The $[X,Y]$-edge colouring game is played with a set of $k$ colours on a graph $G$ with initially uncoloured edges by two players, Alice (A) and Bob (B). The players move alternately. Player $X\in\{A,B\}$ has the first move. $Y\in\{A,B,-\}$. If $Y\in\{A,B\}$, then only player $Y$ may skip any move, otherwise skipping is not allowed for any player. A move consists of colouring an uncoloured edge with one of the $k$ colours such that adjacent edges have distinct colours. When no more moves are possible, the game ends. If every edge is coloured in the end, Alice wins; otherwise, Bob wins. The $[X,Y]$-game chromatic index $\chi_{[X,Y]}'(G)$ is the smallest nonnegative integer $k$ such that Alice has a winning strategy for the $[X,Y]$-edge colouring game played on $G$ with $k$ colours. The graph $G$ is called line $[X,Y]$-perfect if, for any edge-induced subgraph $H$ of $G$, \[\chi_{[X,Y]}'(H)=\omega(L(H)),\] where $\omega(L(H))$ denotes the clique number of the line graph of $H$. For each of the six possibilities $(X,Y)\in\{A,B\}\times\{A,B,-\}$, we characterise line $[X,Y]$-perfect graphs by forbidden (edge-induced) subgraphs and by explicit structural descriptions, respectively.

New Ramsey Multiplicity Bounds and Search Heuristics

AbstractWe study two related problems concerning the number of homogeneous subsets of given size in graphs that go back to questions of Erdős. Most notably, we improve the upper bounds on the Ramsey multiplicity of $$K_4$$ K 4 and $$K_5$$ K 5 and settle the minimum number of independent sets of size 4 in graphs with clique number at most 4. Motivated by the elusiveness of the symmetric Ramsey multiplicity problem, we also introduce an off-diagonal variant and obtain tight results when counting monochromatic $$K_4$$ K 4 or $$K_5$$ K 5 in only one of the colors and triangles in the other. The extremal constructions for each problem turn out to be blow-ups of a graph of constant size and were found through search heuristics. They are complemented by lower bounds established using flag algebras, resulting in a fully computer-assisted approach. For some of our theorems we can also derive that the extremal construction is stable in a very strong sense. More broadly, these problems lead us to the study of the region of possible pairs of clique and independent set densities that can be realized as the limit of some sequence of graphs.

On non-superperfection of edge intersection graphs of paths

The routing and spectrum assignment problem in modern flexgrid elastic optical networks asks for assigning to given demands a route in an optical network and a channel within an optical frequency spectrum so that the channels of two demands are disjoint whenever their routes share a link in the optical network. This problem can be modeled in two phases: firstly, a selection of paths in the network and, secondly, an interval coloring problem in the edge intersection graph of these paths. The interval chromatic number equals the smallest size of a spectrum such that a proper interval coloring is possible, the weighted clique number is a natural lower bound. Graphs where both parameters coincide for all possible non-negative integral weights are called superperfect. Therefore, the occurrence of non-superperfect edge intersection graphs of routing paths can provoke the need of larger spectral resources. In this work, we examine the question which minimal non-superperfect graphs can occur in the edge intersection graphs of routing paths in different underlying networks: when the network is a path, a tree, a cycle, or a sparse planar graph with small maximum degree. We show that for any possible network (even if it is restricted to a path) the resulting edge intersection graphs are not necessarily superperfect. We close with a discussion of possible consequences and of some lines of future research.

The Order Gcd Graph

In this paper we define a simple undirected graph Og(Zn) whose vertices are all the elements of Zn and two distinct vertices a, b are adjacent if and only if gcd (O(a), O(b))= O(a·b) . We introduced that the graph Og(Zn) is complete graph for n=pq . where p, q are distinct prime. The graph Og(Zn) is not planar for n= p2 and n= 2k p. where, p is any odd prime and k > 1 . Also, discuss about Eulerian property of the graph and find degree of vertices and clique number of the graph.

Research Landscape of Filipino Mathematicians: a Co-authorship Network Analysis

Research collaboration presents solutions to various problems that independent researchers might encounter; however, establishing collaboration also poses challenges for most of these researchers. In this study, we created a co-authorship network map of Filipino mathematicians using Elsevier's Scopus database and identified the top five institutions with the largest number of Filipino mathematicians, each with at least one publication in Scopus. We found that among the top three institutions, collaborations rarely occur, with less than 1% distinct inter-institutional collaborations. Various network analysis metrics were performed on the map to establish a baseline for future co-authorship network research. Mathematicians were ranked using both the PageRank algorithm and betweenness centrality to highlight important and bridging figures, respectively. We found that while Canoy SR of Mindanao State University-Iligan Institute of Technology (k = 61), Corcino RB of Cebu Normal University (k = 39), and Mendoza ER of De La Salle University (k = 27) have notable contributions, the University of the Philippines Diliman (UPD) stands out with the most mathematicians having k ≥ 10, where k is the measure of degree centrality. The Leiden community detection algorithm was employed to cluster the network map, revealing 183 distinct clusters – indicating the number of cliques, which are small, exclusive research groups. This highlights the need to strengthen collaborative efforts among Philippine institutions. Interestingly, we found that UPD encompasses a broad range of mathematics research, indicating the potential to serve as a bridge connecting different subject areas. By illustrating the research landscape, this study sheds light on Filipino mathematicians' research, contributing to the enhancement of mathematics research productivity and collaboration in the country.

EDA-Graph: Graph Signal Processing of Electrodermal Activity for Emotional States Detection.

The continuous detection of emotional states has many applications in mental health, marketing, human-computer interaction, and assistive robotics. Electrodermal activity (EDA), a signal modulated by sympathetic nervous system activity, provides continuous insight into emotional states. However, EDA possesses intricate nonstationary and nonlinear characteristics, making the extraction of emotion-relevant information challenging. We propose a novel graph signal processing (GSP) approach to model EDA signals as graphical networks, termed EDA-graph. The GSP leverages graph theory concepts to capture complex relationships in time-series data. To test the usefulness of EDA-graphs to detect emotions, we processed EDA recordings from the CASE emotion dataset using GSP by quantizing and linking values based on the Euclidean distance between the nearest neighbors. From these EDA-graphs, we computed the features of graph analysis, including total load centrality (TLC), total harmonic centrality (THC), number of cliques (GNC), diameter, and graph radius, and compared those features with features obtained using traditional EDA processing techniques. EDA-graph features encompassing TLC, THC, GNC, diameter, and radius demonstrated significant differences (p < 0.05) between five emotional states (Neutral, Amused, Bored, Relaxed, and Scared). Using machine learning models for classifying emotional states evaluated using leave-one-subject-out cross-validation, we achieved a five-class F1 score of up to 0.68.

On zero-divisor graphs of infinite posets

AbstractIt is known that the so-called Beck’s conjecture, i.e. the equality of the finite clique and chromatic numbers of a zero-divisor graph, holds for partially ordered sets Halaš and Jukl (Discrete Math 309(13):4584–4589, 2009). In this paper we present a simple direct proof of this fact. Also, the case when the finiteness assumption of the clique number is omitted is investigated. We have shown that the conjecture fails in general and a bunch of counterexamples is presented.

A weak box-perfect graph theorem

A graph G is called perfect if ω(H)=χ(H) for every induced subgraph H of G, where ω(H) is the clique number of H and χ(H) its chromatic number. The Weak Perfect Graph Theorem of Lovász states that a graph G is perfect if and only if its complement G‾ is perfect. This does not hold for box-perfect graphs, which are the perfect graphs whose stable set polytope is box-totally dual integral.We prove that both G and G‾ are box-perfect if and only if G‾+ is box-perfect, where G+ is obtained by adding a universal vertex to G. Consequently, G+ is box-perfect if and only if G‾+ is box-perfect. As a corollary, we characterize when the complete join of two graphs is box-perfect.

On the Signless Laplacian ABC-Spectral Properties of a Graph

In the paper, we introduce the signless Laplacian ABC-matrix Q̃(G)=D¯(G)+Ã(G), where D¯(G) is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G. The eigenvalues of the matrix Q̃(G) are the signless Laplacian ABC-eigenvalues of G. We give some basic properties of the matrix Q̃(G), which includes relating independence number and clique number with signless Laplacian ABC-eigenvalues. For bipartite graphs, we show that the signless Laplacian ABC-spectrum and the Laplacian ABC-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian ABC-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian ABC-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q̃(G)−tr(Q̃(G))nI, called the signless Laplacian ABC-energy of G. We obtain some upper and lower bounds for signless Laplacian ABC-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the ABC-energy with the signless Laplacian ABC-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian ABC-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same ABC-energy.

Reducing linear Hadwiger’s conjecture to coloring small graphs

In 1943, Hadwiger conjectured that every graph with no K t K_t minor is ( t − 1 ) (t-1) -colorable for every t ≥ 1 t\ge 1 . In the 1980s, Kostochka and Thomason independently proved that every graph with no K t K_t minor has average degree O ( t log ⁡ t ) O(t\sqrt {\log t}) and hence is O ( t log ⁡ t ) O(t\sqrt {\log t}) -colorable. Recently, Norin, Song and the second author showed that every graph with no K t K_t minor is O ( t ( log ⁡ t ) β ) O(t(\log t)^{\beta }) -colorable for every β &gt; 1 / 4 \beta &gt; 1/4 , making the first improvement on the order of magnitude of the O ( t log ⁡ t ) O(t\sqrt {\log t}) bound. The first main result of this paper is that every graph with no K t K_t minor is O ( t log ⁡ log ⁡ t ) O(t\log \log t) -colorable. This is actually a corollary of our main technical result that the chromatic number of a K t K_t -minor-free graph is bounded by O ( t ( 1 + f ( G , t ) ) ) O(t(1+f(G,t))) where f ( G , t ) f(G,t) is the maximum of χ ( H ) a \frac {\chi (H)}{a} over all a ≥ t log ⁡ t a\ge \frac {t}{\sqrt {\log t}} and K a K_a -minor-free subgraphs H H of G G that are small (i.e. O ( a log 4 ⁡ a ) O(a\log ^4 a) vertices). This has a number of interesting corollaries. First as mentioned, using the current best-known bounds on coloring small K t K_t -minor-free graphs, we show that K t K_t -minor-free graphs are O ( t log ⁡ log ⁡ t ) O(t\log \log t) -colorable. Second, it shows that proving Linear Hadwiger’s Conjecture (that K t K_t -minor-free graphs are O ( t ) O(t) -colorable) reduces to proving it for small graphs. Third, we prove that K t K_t -minor-free graphs with clique number at most log ⁡ t / ( log ⁡ log ⁡ t ) 2 \sqrt {\log t}/ (\log \log t)^2 are O ( t ) O(t) -colorable. This implies our final corollary that Linear Hadwiger’s Conjecture holds for K r K_r -free graphs for every fixed r r ; more generally, we show there exists C ≥ 1 C\ge 1 such that for every r ≥ 1 r\ge 1 , there exists t r t_r such that for all t ≥ t r t\ge t_r , every K r K_r -free K t K_t -minor-free graph is C t Ct -colorable. One key to proving the main theorem is building the minor in two new ways according to whether the chromatic number ‘separates’ in the graph: sequentially if the graph is the chromatic-inseparable and recursively if the graph is chromatic-separable. The other key is a new standalone result that every K t K_t -minor-free graph of average degree d = Ω ( t ) d=\Omega (t) has a subgraph on O ( t log 3 ⁡ t ) O(t \log ^3 t) vertices with average degree Ω ( d ) \Omega (d) .

On Three Domination-based Identification Problems in Block Graphs

The problems of determining the minimum-sized identifying, locating-dominating and open locating-dominating codes of an input graph are special search problems that are challenging from both theoretical and computational viewpoints. In these problems, one selects a dominating set C of a graph G such that the vertices of a chosen subset of V(G) (i.e. either V(G) \ C or V(G) itself) are uniquely determined by their neighborhoods in C. A typical line of attack for these problems is to determine tight bounds for the minimum codes in various graph classes. In this work, we present tight lower and upper bounds for all three types of codes for block graphs (i.e. diamond-free chordal graphs). Our bounds are in terms of the number of maximal cliques (or blocks) of a block graph and the order of the graph. Two of our upper bounds verify conjectures from the literature with one of them being now proven for block graphs in this article. As for the lower bounds, we prove them to be linear in terms of both the number of blocks and the order of the block graph. We provide examples of families of block graphs whose minimum codes attain these bounds, thus showing each bound to be tight.

A Conjecture for the Clique Number of Graphs Associated with Symmetric Numerical Semigroups of Arbitrary Multiplicity and Embedding Dimension

A subset S of non-negative integers No is called a numerical semigroup if it is a submonoid of No and has a finite complement in No. An undirected graph G(S) associated with S is a graph having V(G(S))={vi:i∈No∖S} and E(G(S))={vivj⇔i+j∈S}. In this article, we propose a conjecture for the clique number of graphs associated with a symmetric family of numerical semigroups of arbitrary multiplicity and embedding dimension. Furthermore, we prove this conjecture for the case of arbitrary multiplicity and embedding dimension 7.

Extended total graph associated to finite commutative rings

UDC 512.5 For a commutative ring R with nonzero identity 1 ≠ 0 , let Z ( R ) denote the set of zero divisors. The total graph of R denoted by T Γ ( R ) is a simple graph in which all elements of R are vertices and any two distinct vertices x and y are adjacent if and only if x + y ∈ Z ( R ) . In this paper, we define an extension of the total graph denoted by T ( Γ e ( R ) ) with vertex set as Z ( R ) , and two distinct vertices x and y are adjacent if and only if x + y ∈ Z * ( R ) , where Z * ( R ) is the set of nonzero zero divisors of R . Our main aim is to characterize the finite commutative rings whose T ( Γ e ( R ) ) has clique numbers 1,2 , and 3 . In addition, we characterize finite commutative nonlocal rings R for which the corresponding graph T ( Γ e ( R ) ) has the clique number 4.

Cliques in squares of graphs with maximum average degree less than 4

AbstractHocquard, Kim, and Pierron constructed, for every even integer , a 2‐degenerate graph with maximum degree such that . We prove for (a) all 2‐degenerate graphs and (b) all graphs with , upper bounds on the clique number of that match the lower bound given by this construction, up to small additive constants. We show that if is 2‐degenerate with maximum degree , then (with when is sufficiently large). And if has and maximum degree , then . Thus, the construction of Hocquard et al. is essentially the best possible. Our proofs introduce a “token passing” technique to derive crucial information about nonadjacencies in of vertices that are adjacent in . This is a powerful technique for working with such graphs that has not previously appeared in the literature.

Quantum circuits for discrete graphical models

Graphical models are useful tools for describing structured high-dimensional probability distributions. The development of efficient algorithms for generating samples thereof remains an active research topic. In this work, we provide a quantum algorithm that enables the generation of unbiased and independent samples from general discrete graphical models. To this end, we identify a coherent embedding of the graphical model based on a repeat-until-success sampling scheme which clearly identifies whether a drawn sample represents the underlying distribution. Furthermore, we show that the success probability for finding a valid sample can be lower bounded with a quantity that depends on the number of maximal cliques and the model parameter norm. Moreover, we rigorously proof that the quantum embedding conserves the key property of graphical models, i.e., factorization over the cliques of the underlying conditional independence structure. The quantum sampling algorithm also allows for maximum likelihood learning as well as maximum a posteriori state approximation for the graphical model. Finally, the proposed quantum method shows potential to realize interesting problems on near-term quantum processors. In fact, illustrative experiments demonstrate that our method can carry out sampling and parameter learning not only with idealized simulations of quantum computers but also on actual quantum hardware solely supported by simple readout error mitigation.

Ant algorithms for finding weighted and unweighted maximum cliques in d-division graphs

This article deals with the problem of finding the maximum number of maximum cliques in a weighted graph with all edges between vertices from different d-division of a graph with the minimum total weight of all these cliques, and the problem of finding the maximum number of maximum cliques in a nonweighted graph with not all edges between vertices from different d-division of the graph. This article presents new ant algorithms with new desire functions for these problems. These algorithms were tested for their purpose with different changing input parameters, the test results were tabulated and discussed, the best algorithms were indicated.

Ramsey‐type problems on induced covers and induced partitions toward the Gyárfás–Sumner conjecture

AbstractGyárfás and Sumner independently conjectured that for every tree , there exists a function such that every ‐free graph satisfies , where and are the chromatic number and the clique number of , respectively. This conjecture gives a solution of a Ramsey‐type problem on the chromatic number. For a graph , the induced SP‐cover number (resp. the induced SP‐partition number ) of is the minimum cardinality of a family of induced subgraphs of such that each element of is a star or a path and (resp. ). Such two invariants are directly related concepts to the chromatic number. From the viewpoint of this fact, we focus on Ramsey‐type problems for two invariants and , which are analogies of the Gyárfás‐Sumner conjecture, and settle them. As a corollary of our results, we also settle other Ramsey‐type problems for widely studied invariants.