Clique Number Research Articles

Recognition and Optimization Algorithms for P5-Free Graphs

The weighted independent set problem on P 5 -free graphs has numerous applications, including data mining and dispatching in railways. The recognition of P 5 -free graphs is executed in polynomial time. Many problems, such as chromatic number and dominating set, are NP-hard in the class of P 5 -free graphs. The size of a minimum independent feedback vertex set that belongs to a P 5 -free graph with n vertices can be computed in O ( n 16 ) time. The unweighted problems, clique and clique cover, are NP-complete and the independent set is polynomial. In this work, the P 5 -free graphs using the weak decomposition are characterized, as is the dominating clique, and they are given an O ( n ( n + m ) ) recognition algorithm. Additionally, we calculate directly the clique number and the chromatic number; determine in O ( n ) time, the size of a minimum independent feedback vertex set; and determine in O ( n + m ) time the number of stability, the dominating number and the minimum clique cover.

Some properties of intersection graph of a module with an application of the graph of ℤn

Let R be an unital ring which is not necessarily commutative. The intersection graph of ideals of R is a graph with the vertex set which contains proper ideals of R and distinct two vertices I and J are adjacent if and only if I Ç J ¹ 0 is denoted by G. In this paper, we will give some properties of regular graph, triangle-free graph and clique number of G(M) for a module M. We also characterize girth of an Artinian module with connected module. We characterize the chromatic number of G(Z n ). We also give an algorithm for the chromatic number of G(Z n ).

On the chromatic number of disjointness graphs of curves

Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis.We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that ω(G)=k, then χ(G)≤(k+12). If we only require that every curve is x-monotone and intersects the y-axis, then we have χ(G)≤k+12(k+23). Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist Kk-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Ω(k4) colors. This matches the upper bound up to a constant factor.

Maximizing the density of [formula omitted]’s in graphs of bounded degree and clique number

Zykov showed in 1949 that among graphs on n vertices with clique number ω(G)≤ω, the Turán graph Tω(n) maximizes not only the number of edges but also the number of copies of Kt for each size t. The problem of maximizing the number of copies of Kt has also been studied within other classes of graphs, such as those on n vertices with maximum degree Δ(G)≤Δ.We combine these restrictions and investigate which graphs with Δ(G)≤Δ and ω(G)≤ω maximize the number of copies of Kt per vertex. We define ft(Δ,ω) as the supremum of ρt, the number of copies of Kt per vertex, among such graphs, and show for fixed t and ω that ft(Δ,ω)=(1+o(1))ρt(Tω(Δ+⌊Δω−1⌋)). For two infinite families of pairs (Δ,ω), we determine ft(Δ,ω) exactly for all t≥3. For another we determine ft(Δ,ω) exactly for the two largest possible clique sizes. Finally, we demonstrate that not every pair (Δ,ω) has an extremal graph that simultaneously maximizes the number of copies of Kt per vertex for every size t.

Combinatorics of $k$-Farey graphs

With an eye towards studying curve systems on low-complexity surfaces, we introduce and analyze the k-Farey graphs ℱk and ℱ≤k, two natural variants of the Farey graph ℱ in which we relax the edge condition to indicate intersection number =k or ≤k, respectively. The former, ℱk, is disconnected when k>1. In fact, we find that the number of connected components is infinite if and only if k is not a prime power. Moreover, we find that each component of ℱk is a quasitree (in fact, a tree when k is even) and Aut(ℱk) is uncountable for k>1. As for ℱ≤k, Agol obtained an upper bound of 1+ min{p:p is a prime>k} for both chromatic and clique numbers, and observed that this is an equality when k is either one or two less than a prime. We add to this list the values of k that are three less than a prime equivalent to 11(mod12), and we show computer-assisted computations of many values of k for which equality fails.

A Systematic Approach to Group Properties Using its Geometric Structure

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

A Systematic Approach to Group Properties Using its Geometric Structure

The algebraic properties of a group can be explored through the relationship among its elements. In this paper, we define the graph that establishes a systematic relationship among the group elements. Let G be a finite group, the order product prime graph of a group G, is a graph having the elements of G as its vertices and two vertices are adjacent if and only if the product of their order is a prime power. We give the general presentation for the graph on dihedral groups and cyclic groups and classify finite dihedral groups and cyclic groups in terms of the order product prime graphs as one of connected, complete, regular and planar. We also obtained some invariants of the graph such as its diameter, girth,independent number and the clique number. Furthermore, we used thevertex-cut of the graph in determining the nilpotency status of dihedralgroups. The graph on dihedral groups is proven to be regular and complete only if the degree of the corresponding group is even prime power and connected for all prime power degree. It is also proven on cyclic groups to be both regular, complete and connected if the group has prime power order. Additionally, the result turn out to show that any dihedral group whose order product prime graph’s vertex-cut is greater than one is nilpotent. We also show that the order product prime graph is planar only when the degree of the group is three for dihedral groups and less than five for cyclic groups. Our final result shows that the order product prime graphs of any two isomorphic groups are isomophic.

On clique‐inverse graphs of graphs with bounded clique number

AbstractThe clique graph K(G) of G is the intersection graph of the family of maximal cliques of G. For a family of graphs, the family of clique‐inverse graphs of , denoted by , is defined as . Let be the family of Kp‐free graphs, that is, graphs with clique number at most p − 1, for an integer constant p ≥ 2. Deciding whether a graph H is a clique‐inverse graph of can be done in polynomial time; in addition, for can be characterized by a finite family of forbidden induced subgraphs. In Protti and Szwarcfiter, the authors propose to extend such characterizations to higher values of p. Then a natural question arises: Is there a characterization of by means of a finite family of forbidden induced subgraphs, for any p ≥ 2? In this note we give a positive answer to this question. We present upper bounds for the order, the clique number, and the stability number of every forbidden induced subgraph for in terms of p.

Induced subgraphs of graphs with large chromatic number. VI. Banana trees

We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper [6], the first author proved that every tree has this property; and in another earlier paper with Maria Chudnovsky [2], we proved that every cycle has this property. Here we give a common generalization. Say a “banana” is the union of a set of paths all with the same ends but otherwise disjoint. We prove that if H is obtained from a tree by replacing each edge by a banana then H has the property mentioned.

Anzahl theorems of matrices over the residue class ring modulo psqt and their applications

Let h=psqt be its decomposition into a product of powers of two distinct primes, and Zh be the residue class ring modulo h. In this paper, we determine the Smith normal forms of matrices, compute the number of the orbits of m×n matrices under the action of the group GLm(Zh)×GLn(Zh) and the length of each orbit over Zh. As applications, we determine the clique number, the independence number and the chromatic number of the bilinear forms graph, construct a family of association schemes by using 1×n matrices and compute the intersection numbers and the character table over Zh for s=t=1.

A homogeneous polynomial associated with general hypergraphs and its applications

In this paper, we define a homogeneous polynomial for a general hypergraph, and establish a remarkable connection between clique number and the homogeneous polynomial of a general hypergraph. For a general hypergraph, we explore some inequality relations among spectral radius, clique number and the homogeneous polynomial. We also give lower and upper bounds on the spectral radius of a general hypergraph in terms of the clique number.

Some Bounds on Zeroth-Order General Randić Index

For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D ( G ) = ∑ u ∈ V ( G ) d ( u ) − 1 where d ( u ) is the number of vertices adjacent to the vertex u in G. By replacing − 1 by any non-zero real number we obtain zeroth-order general Randić index, i.e., 0 R γ ( G ) = ∑ u ∈ V ( G ) d ( u ) γ , where γ ∈ R − { 0 } . Xu et al. investigated some lower and upper bounds on I D for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for γ < 0 . The corresponding extremal graphs have also been identified.

On the Number of Cliques in Graphs with a Forbidden Subdivision or Immersion

How many cliques can a graph on $n$ vertices have with a forbidden substructure? Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of ...

Adjacency matrix and eigenvalues of the zero divisor graph Γ(Zn)

Let R be a commutative ring with non-zero identity and Z^∗(R) be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices x, y ∈ Z^∗(R) are adjacent if and only if xy = 0. In this paper, the eigenvalues of Γ(Zn) for n = p^2q^2, where p and q are distinct primes, are investigated. Also, the girth, diameter, clique number and stability number of this graph are found.

Enumerating cliques in direct product graphs

The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two vertices are adjacent if and only if their difference is relatively prime to $n$. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erdős and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order $m$ in the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula involves an exciting class of arithmetic functions known as Schemmel totient functions, which we also briefly discuss. More generally, the proof yields a formula for the number of cliques of order $m$ in a direct product of balanced complete multipartite graphs.

Impact of higher order network structure on emergent cortical activity.

Synaptic connectivity between neocortical neurons is highly structured. The network structure of synaptic connectivity includes first-order properties that can be described by pairwise statistics, such as strengths of connections between different neuron types and distance-dependent connectivity, and higher order properties, such as an abundance of cliques of all-to-all connected neurons. The relative impact of first- and higher order structure on emergent cortical network activity is unknown. Here, we compare network structure and emergent activity in two neocortical microcircuit models with different synaptic connectivity. Both models have a similar first-order structure, but only one model includes higher order structure arising from morphological diversity within neuronal types. We find that such morphological diversity leads to more heterogeneous degree distributions, increases the number of cliques, and contributes to a small-world topology. The increase in higher order network structure is accompanied by more nuanced changes in neuronal firing patterns, such as an increased dependence of pairwise correlations on the positions of neurons in cliques. Our study shows that circuit models with very similar first-order structure of synaptic connectivity can have a drastically different higher order network structure, and suggests that the higher order structure imposed by morphological diversity within neuronal types has an impact on emergent cortical activity.

On zero divisor graphs of the rings $$Z_n$$

For a commutative ring R with non-zero zero-divisor set $$Z^*(R)$$, the zero-divisor graph of R is $$\varGamma (R)$$ with vertex set $$Z^*(R)$$, where two distinct vertices x and y are adjacent if and only if $$xy=0$$. The zero-divisor graph structure of $${\mathbb {Z}}_{p^n}$$ is described. We determine the clique number, degree of the vertices, size, metric dimension, upper dimension, automorphism group, Wiener index of the associated zero-divisor graph of $${\mathbb {Z}}_{p^n}$$. Further, we provide a partition of the vertex set of $$\varGamma ({\mathbb {Z}}_{p^n})$$ into distance similar equivalence classes and we show that in this graph the upper dimension equals the metric dimension. Also, we discuss similar properties of the compressed zero-divisor graph.

On the edge metric dimension of graphs

Let $G = (V, E)$ be a connected graph of order $n$. $S \subseteq V$ is an edge metric generator of $G$ if any pair of edges in $E$ can be distinguished by some element of $S$. The edge metric dimension $edim(G)$ of a graph $G$ is the least size of an edge metric generator of $G$. In this paper, we give the characterization of all connected bipartite graphs with $edim = n-2$, which partially answers an open problem of Zubrilina (2018). Furthermore, we also give a sufficient and necessary condition for $edim(G) = n-2$, where $G$ is a graph with maximum degree $n-1$. In addition, the relationship between the edge metric dimension and the clique number of a graph $G$ is investigated by construction.

On the sum of signless Laplacian spectra of graphs

For a simple graph $G(V,E)$ with $n$ vertices, $m$ edges, vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$ and edge set $E(G)=\{e_1, e_2,\dots, e_m\}$, the adjacency matrix $A=(a_{ij})$ of $G$ is a $(0, 1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if $v_i$ is adjacent to $v_j$ and equal to 0, otherwise. Let $D(G)={diag}(d_1, d_2, \dots, d_n)$ be the diagonal matrix associated to $G$, where $d_i=\deg(v_i),$ for all $i\in \{1,2,\dots,n\}$. The matrices $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum ($L$-spectrum) and the signless Laplacian spectrum ($Q$-spectrum) of the graph $G$. If $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$ are the Laplacian eigenvalues of $G$, Brouwer conjectured that the sum of $k$ largest Laplacian eigenvalues $S_{k}(G)$ satisfies $S_{k}(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}$ and this conjecture is still open. If $q_1,q_2, \dots, q_n$ are the signless Laplacian eigenvalues of $G$, for $1\leq k\leq n$, let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of $k$ largest signless Laplacian eigenvalues of $G$. Analogous to Brouwer's conjecture, Ashraf et al. conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $1\leq k\leq n$. This conjecture has been verified in affirmative for some classes of graphs. We obtain the upper bounds for $S^{+}_{k}(G)$ in terms of the clique number $\omega$, the vertex covering number $\tau$ and the diameter of the graph $G$. Finally, we show that the conjecture holds for large families of graphs.

The Radial Radio Number and the Clique Number of a Graph

Let )) G(V(G),E(G be a graph. A radial radio labeling, f, of a connected graph G is an assignment of positive integers to the vertices satisfying the following condition: d (u, v)+ | f (u) − f (v) | 1 + r(G) , for any two distinct vertices ) ( , GVv u  , where ) , ( v u d and )(Gr denote the distance between the vertices u and v and the radius of the graph G, respectively. The span of a radial radio labeling f is the largest integer in the range of f and is denoted by span(f). The radial radio number of G, r(G) , is the minimum span taken over all radial radio labelingsof G. In this paper, we construct a graph a graph for which the difference between the radial radio number and the clique number is the given non negative integer.