We prove necessary conditions for certain elementary symmetric functions, e λ , to appear with nonzero coefficient in Stanley’s chromatic symmetric function as well as in the generalization considered by Shareshian and Wachs. We do this by first considering the expansion in the monomial or Schur basis and then performing a basis change. Using the former, we make a connection with two fundamental graph theory invariants, the independence and clique numbers. This allows us to prove nonnegativity of three-column coefficients for all natural unit interval graphs, giving more insight into the Stanley–Stembridge Conjecture, recently proven by Hikita, and the Shareshian–Wachs Conjecture. The Schur basis permits us to give a new interpretation of the coefficient of e n in terms of tableaux. We are also able to give an explicit formula for that coefficient.

The maximum number of cliques in graphs that avoid vertex-disjoint copies of path of length two

In this paper, we determine the unique graph whose least eigenvalue attains the minimum among all the connected simple graphs with given clique number.

Let [Formula: see text] and [Formula: see text] respectively denote the chromatic number and clique number of a graph [Formula: see text]. A [Formula: see text] is a path on 5 vertices, a banner (paw) is the graph obtained by joining a new vertex to a single vertex of [Formula: see text] ([Formula: see text]) and a hammer is obtained by subdividing the pendant edge of a paw exactly once. Recently, [Formula: see text]-free graphs have received wide attention. In 2019, Karthick, Maffray and Pastor, gave a structural characterisation of [Formula: see text]-free graphs, which when combined with a result by Bourneuf and Thomassé (2023) implies that for a [Formula: see text]-free graph [Formula: see text], [Formula: see text]. Geißer in his thesis (2022) showed that the [Formula: see text]-binding function of the class of [Formula: see text]-free graphs is bounded by the [Formula: see text]-binding function of [Formula: see text]-free graphs. By a result of Kim (1995), the chromatic number [Formula: see text] of a [Formula: see text]-free graph [Formula: see text] has order of magnitude [Formula: see text]. Recently, Song and Xu (2024) proved that every ([Formula: see text], [Formula: see text], banner, hammer)-free graph [Formula: see text] is [Formula: see text]-colorable. This motivates us to study the subclasses of ([Formula: see text], banner)-free graphs. We prove that for any ([Formula: see text], banner, [Formula: see text])-free graph [Formula: see text] where [Formula: see text], [Formula: see text] for [Formula: see text]. Moreover, the bound is tight for [Formula: see text].

A tensor's spectral bound on the clique number

The anti-Ramsey numbers of cliques in complete multi-partite graphs

This study presents a structural investigation of the Co-Intersection graph, denoted by [Formula: see text], which is defined on the set of all non-trivial ideals of a commutative ring [Formula: see text]. In this undirected graph, two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever the sum of the corresponding ideals satisfies [Formula: see text]. Although co-intersection graphs were previously introduced and examined by Hoseini and Talebi [The co-intersection graphs of ideals of rings, Ann. Pure Appl. Math. 28(2) (2023) 63–71], the existing literature does not provide a classification of finite commutative rings according to the clique number of [Formula: see text]. This work fills this gap by completely characterizing all finite commutative rings [Formula: see text] for which the clique number of [Formula: see text]. Furthermore, explicit formulas are derived for the first and second Zagreb indices of [Formula: see text], where [Formula: see text] is a prime and [Formula: see text], thereby contributing new topological invariants to the study of co-intersection graphs associated with principal ideal rings. These results expand earlier studies and explain more clearly how properties of a ring are connected to the features of its co-intersection graph.

Abstract We study cliques in graphs arising from quadratic forms where the vertices are the elements of the module of the quadratic form and two vertices are adjacent if their difference represents some fixed scalar. We determine structural properties and the clique number for quadratic forms over finite rings. We further extend previous results about graphs arising from such forms and forms over fields of characteristic 0 in a unified framework.

The codegree codeg ( 𝒫 ) of a lattice polytope 𝒫 is a fundamental invariant in discrete geometry. In the present paper, we investigate the codegree of the stable set polytope 𝒫 G associated with a simple graph G . Specifically, we establish the inequalities ω ( G ) + 1 ≤ codeg ( 𝒫 G ) ≤ χ ( G ) + 1 , where ω ( G ) and χ ( G ) denote the clique number and the chromatic number of G , respectively. Furthermore, an explicit formula for codeg ( 𝒫 G ) is given when G is either a line graph or an h -perfect graph. Finally, as an application of these results, we provide upper and lower bounds on the regularity of the toric ring associated with 𝒫 G .

Let R be a finite commutative ring with identity 1. The U ‐clean graph U ‐ C l ( R ) of a ring R is a simple undirected graph with vertices are of the form ( e , u ), where e is a nonzero idempotent and u is a unit of R , and two distinct vertices ( e , u ), ( f , v ) of U ‐ C l ( R ) are adjacent if and only if e = f = 1 or u v = 1. In this paper, we present the strong resolving graph U ‐ C l ( R ) S R of U ‐ C l ( R ). The vertex degrees, the independence number, the clique number, and the chromatic number of U ‐ C l ( R ) S R are determined. Moreover, we show that if k is a divisor of ( n − 1) n , where k and n are positive integers with k > n ≥ 3, then U ‐ is not a complete graph. Finally, we give some examples of U ‐ and U ‐ to illustrate our results.

Let G(Γ,Ε) be a simple, connected and undirected graph. A radio labeling of G, ψ:Γ→{1,2,3,…} is a function satisfying the condition for any two distinct vertices u and v, d(u,v)+|ψ(u)-ψ(v)|≥1+diam(G), where d(u,v) denotes the distance between the vertices u and v and diam(G) is the diameter of the graph G. The span of a radio labeling is the maximum integer that assigns to a vertex and radio number, rn(G) is the minimum span taken overall radio labelings of G. This paper presents relationships between the radio number clique number and chromatic number of a simple connected graph.

The maximum number of cliques in graphs with given fractional matching number and minimum degree

Let be a commutative ring with , and be an -module. The minimal intersection graph of , denoted by is a simple undirected graph whose vertices are proper non-zero submodules of and any two distinct vertices and are adjacent if and only if be an minimal (= simple) submodule of . In this article, we explore connectedness, clique number, split character, planarity, independence number and domination number of .

Abstract Alon and Shikhelman initiated the systematic study of a generalization of the extremal function. Motivated by algorithmic applications, the study of the extremal function , that is, the number of cliques of order in ‐minor free graphs on vertices, has received much attention. In this paper, we determine essentially sharp bounds on the maximum possible number of cliques of order in a ‐minor free graph on vertices. More precisely, we determine a function such that for each with , every ‐minor free graph on vertices has at most cliques of order . We also show this bound is sharp by constructing a ‐minor‐free graph on vertices with cliques of order . This bound answers a question of Wood and Fox–Wei asymptotically up to in the exponent except the extreme values when is very close to .

The number of edges in graphs with bounded clique number and circumference

Extremal graphs for star forests with bounded clique number

We investigate the clique numbers and structural properties of commuting graphs associated with direct sum matrix rings over finite commutative rings. For a finite commutative ring L with unity, we study the commuting graph \(\Gamma(M(m \oplus m, L))\) whose vertex set consists of all non-central matrices in \(M(m \oplus m, L)\), where two distinct vertices are adjacent if and only if they commute. Our main contributions establish fundamental lower bounds for the clique number \(\omega\Gamma(M(m \oplus m, L)))\) across various ring structures. We prove that for any finite commutative ring R with unity and positive integer \(m \geq 3\), the clique number satisfies \(\omega(\Gamma(M(m, R))) \geq |R|^{2m} - |R|^2\). For rings isomorphic to \({Z}_{p^r}\) where \(r \geq 3\) is odd, we establish the improved bound \(\omega(\Gamma(M(m, R))) \geq \max\{(p^r)^{2m} - p^{2r}, (p^{r-1})^{m^2-m}(p^{r+1})^{m-1}p^{2r} - p^{2r}\}\). When \(r \geq 2\) is even, the bound becomes \(\omega(\Gamma(M(m, R))) \geq \max\{(p^r)^{2m} - p^{2r}, (p^r)^{m^2-1}p^{2r} - p^{2r}\}\). Our approach combines sophisticated matrix-theoretic techniques with graph-theoretic analysis to construct explicit maximal cliques and derive optimal bounds. The results provide new insights into the intersection of algebraic graph theory and matrix ring theory, with potential applications in coding theory and combinatorial optimization.

A graph G of order n is called Diophantine if there exists a labeling function f of vertices such that gcd(f(u),f(v)) divides n for every pair adjacent vertices u,v in G. This paper defines, studies and generalizes maximal Diophantine graphs Dn, determining their independence number, number of full-degree vertices, and clique number. These parameters establish necessary conditions for the existence of Diophantine labelings.

Abstract Given a commutative ring R R with unity, and an endomorphism f f on R R , the graph G ( R , f ) G\left(R,f) assigned to R R is a simple graph with vertex set R R , and two distinct vertices r r and s s are adjacent if r f ( s ) = 0 rf\left(s)=0 or s f ( r ) = 0 sf\left(r)=0 . This graph generalizes Beck’s zero-divisor graph G ( R ) G\left(R) . This extension enables a deeper insight into both the algebraic structure and graph-theoretic properties. We explore the properties of G ( R , f ) G\left(R,f) , such as connectivity, completeness, and cycles. We also determine the values for the diameter, girth, independence number, clique number, chromatic number, and domination number, sometimes under algebraic conditions on R R or f f . The study also reveals how algebraic properties of R R and f f relate to the graph-theoretic properties of G ( R , f ) G\left(R,f) . Applications to the ring Z n {{\mathbb{Z}}}_{n} are provided with illustrative examples. The ties between G ( R , f ) G\left(R,f) and Beck’s graph G ( R ) G\left(R) are also presented offering new methods to tackle problems in the study of zero-divisor graphs.

We introduce multiscale topological analysis for studying cryptocurrency price series in the time domain. This is achieved by first performing a coarse-grained procedure on the volatility series at multiple temporal scales, and then constructing consecutive visibility graphs from the resulting coarse-grained series. We show that their degree distribution presents a likely power-law behavior. This scaling characteristics keeps invariant even varying time scale factor. Interestingly, we find that the number of cliques that capturing higher-order relations, presents a clear power-law behavior with the time scale factor. Their associated scaling exponent shows a monotonically decreasing pattern. Our work reveals the function of higher-order topological structure underlying cryptocurrency time series.