We obtain explicit expressions for the class in the Grothendieck group of varieties of the moduli space $\overline {\mathcal{M}}_{0,n}$. This information is equivalent to the Poincaré polynomial and yields explicit expressions for the Betti numbers of $\overline {\mathcal{M}}_{0,n}$ in terms of Stirling or Bernoulli numbers. The expressions are obtained by solving a differential equation characterizing the generating function for the Poincaré polynomials, determined by Manin in the 1990s and equivalent to Keel’s recursion for the Betti numbers of $\overline {\mathcal{M}}_{0,n}$. Our proof reduces the solution to two combinatorial identities, verified by applying Lagrange series. We also study generating functions for the individual Betti numbers. These functions are determined by a set of polynomials $p^{(k)}_m(z)$, $k\geqslant m$. These polynomials are conjecturally log-concave; we verify this conjecture for several infinite families, corresponding to generating functions for $2k$-Betti numbers of $\overline {\mathcal{M}}_{0,n}$ for all $k\leqslant 100$. Further, studying the polynomials $p^{(k)}_m(z)$, we prove that the generating function for the Grothendieck class can be written in terms of a series of rational functions in the principal branch of the Lambert W-function. We include an interpretation of the main result in terms of Stirling matrices and a discussion of the Euler characteristic of $\overline {\mathcal{M}}_{0,n}$.

We prove that the family of graphs containing no cycle with exactly k-chords is $\chi $-bounded, for k large enough or of form $\ell (\ell -2)$ with $\ell \ge 3$ an integer. This verifies (up to a finite number of values k) a conjecture of Aboulker and Bousquet (2015).

We consider a class of sequence $c(n)$ with the following asymptotic form:

$$ \begin{align*}c(n) \sim \frac C{n^\kappa} \exp\left(\sum_{\lambda\in\mathcal S}A_\lambda n^\lambda \right) \sum_{\mu\in\mathcal T} \frac{\beta_\mu}{n^\mu} \qquad (n \to \infty).\end{align*} $$

$c(n)$

$c(n)$

$\ell $

$S_n$

$\mathfrak {su}(3)$

$\mathfrak {so}(5),$

A complete classification of unimodular valuations on the set of lattice polygons with values in the spaces of polynomials and formal power series, respectively, is established. The valuations are classified in terms of their behavior with respect to dilation using extensions to unbounded polyhedra and basic invariant theory.

Let ${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of $\mathrm {GL}_{n}$, where $a+b+c+d=n$. In this paper, we study the class of all pattern subgroups of ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$. We establish character-number formulae of degree $q^e$ for all these pattern groups. For pattern subgroups $G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.

Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial t-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. We assume here that we have a commutative association scheme with irrational eigenvalues and wish to study its Delsarte T-designs. We explore when a T-design is also a $T'$-design, where $T'\supseteq T$ is controlled by the orbits of a Galois group related to the splitting field of the association scheme. We then study Delsarte designs in the association schemes of finite groups, with a detailed exploration of the dicyclic groups.

Let $X_H$ be the number of copies of a fixed graph H in G(n,p). In 2016, Gilmer and Kopparty conjectured that a local central limit theorem should hold for $X_H$ as long as H is connected, $p\gg n^{-1/m(H)}$ and $n^2(1-p)\gg 1$, where m(H) denotes the m-density of H. Recently, Sah and Sawhney showed that the Gilmer–Kopparty conjecture holds for constant p. In this paper, we show that the Gilmer–Kopparty conjecture holds for triangle counts in the sparse range. More precisely, if $p \in (4n^{-1/2}, 1/2)$, then

\[ {} {} {}\sup_{x\in \mathcal{L}}\left| \dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}-\sigma\cdot \mathbb{P}(X^* = x)\right|=n^{-1/2+o(1)}p^{1/2}, {} {}\]

$\sigma^2 = \mathbb{V}\text{ar}(X_{K_3})$

$X^{*}=(X_{K_3}-\mathbb{E}(X_{K_3}))/\sigma$

$\mathcal{L}$

$X^*$

$n^{-1}\ll p \lt c$

$c\in (0,1)$

$\ell_1$

$m_2$

A meta-conjecture of Coulson, Keevash, Perarnau, and Yepremyan [12] states that above the extremal threshold for a given spanning structure in a (hyper-)graph, one can find a rainbow version of that spanning structure in any suitably bounded colouring of the host (hyper-)graph. We solve one of the most pertinent outstanding cases of this conjecture by showing that for any $1\leq j\leq k-1$, if $G$ is a $k$-uniform hypergraph above the $j$-degree threshold for a loose Hamilton cycle, then any globally bounded colouring of $G$ contains a rainbow loose Hamilton cycle.

Let $\mathcal{G}$ be the class of all connected simple graphs. The Hoffman program of graphs with respect to a spectral invariant $\lambda(G)$ consists of determining all the limit points of the set $\{\lambda(G)\,\vert\, G\in\mathcal{G}\}$ and characterising all $G$’s such that $\lambda(G)$ does not exceed a fixed limit point. In this paper, we study the Hoffman program for Laplacian matching polynomials of graphs in regard to their largest Laplacian matching roots. Precisely, we determine all the limit points of the largest Laplacian matching roots of graphs less than $\tau = 2+\omega^{\frac{1}{2}}+\omega^{-\frac{1}{2}}(=4.38+)$, and then characterise the connected graphs with the largest Laplacian matching roots less than $2+\sqrt{5}$, where $\omega=\frac{1}{3}(\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}+1)$.

We consider the problem of detecting whether a power-law inhomogeneous random graph contains a geometric community, and we frame this as a hypothesis-testing problem. More precisely, we assume that we are given a sample from an unknown distribution on the space of graphs on n vertices. Under the null hypothesis, the sample originates from the inhomogeneous random graph with a heavy-tailed degree sequence. Under the alternative hypothesis, $k=o(n)$ vertices are given spatial locations and connect following the geometric inhomogeneous random graph connection rule. The remaining $n-k$ vertices follow the inhomogeneous random graph connection rule. We propose a simple and efficient test based on counting normalized triangles to differentiate between the two hypotheses. We prove that our test correctly detects the presence of the community with high probability as $n\to\infty$, and identifies large-degree vertices of the community with high probability.

The main goal of this paper is to introduce a new model of evolvement of beliefs on networks. It generalizes the DeGroot model and describes the iterative process of establishing the consensus in isolated social networks in the case of nonlinear aggregation functions. Our main tools come from mean theory and graph theory. The case, when the root set of the network (influencers, news agencies, etc.) is ergodic is fully discussed. The other possibility, when the root contains more than one component, is partially discussed and it could be a motivation for further research.

In this article, we present a unified approach for proving several Turán-type and generalized Turán-type problems, degree power problems, and extremal spectra problems on paths, cycles, and matchings. Specifically, we generalize classical results on cycles and matchings by Kopylov, Erdős–Gallai, and Luo et al., respectively, and provide a positive resolution to an open problem originally proposed by Nikiforov. Moreover, we improve the spectral extremal results on paths, building on the work of Nikiforov, and Nikiforov and Yuan. Additionally, we provide a comprehensive solution to the connected version of the problem related to the degree power sum of a graph that contains no path on k vertices, a topic initially investigated by Caro and Yuster.

The generalised random graph contains n vertices with positive i.i.d. weights. The probability of adding an edge between two vertices is increasing in their weights. We require the weight distribution to have finite second moments, and study the point process $\mathcal{C}_n$ on $\{3,4,\ldots\}$, which counts how many cycles of the respective length are present in the graph. We establish convergence of $\mathcal{C}_n$ to a Poisson point process. Under the stronger assumption of the weights having finite fourth moments we provide the following results. When $\mathcal{C}_n$ is evaluated on a bounded set A, we provide a rate of convergence. If the graph is additionally subcritical, we extend this to unbounded sets A at the cost of a slower rate of convergence. From this we deduce the limiting distribution of the length of the shortest and longest cycles when the graph is subcritical, including rates of convergence. All mentioned results also apply to the Chung–Lu model and the Norros–Reittu model.

We study perfect codes in the homogeneous metric over the ring $\mathbb {Z}_{{2}^k}, k\ge 2$. We derive arithmetic nonexistence results from Diophantine equations on the parameters resulting from the sphere packing conditions, and a Lloyd theorem based on the theory of weakly metric association schemes.

We prove that on any transitive graph G with infinitely many ends, a self-avoiding walk of length n is ballistic with extremely high probability, in the sense that there exist constants $c,t>0$ such that $\mathbb {P}_n(d_G(w_0,w_n)\geq cn)\geq 1-e^{-tn}$ for every $n\geq 1$. Furthermore, we show that the number of self-avoiding walks of length n grows asymptotically like $\mu _w^n$, in the sense that there exists $C>0$ such that $\mu _w^n\leq c_n\leq C\mu _w^n$ for every $n\geq 1$. These results generalise earlier work by Li (J. Comb. Theory Ser. A, 2020). The key to this greater generality is that in contrast to Li’s approach, our proof does not require the existence of a special structure that enables the construction of separating patterns. Our results also extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of G which does not fix an end of G.

The Pósa–Seymour conjecture determines the minimum degree threshold for forcing the $k$th power of a Hamilton cycle in a graph. After numerous partial results, Komlós, Sárközy, and Szemerédi proved the conjecture for sufficiently large graphs. In this paper, we focus on the analogous problem for digraphs and for oriented graphs. We asymptotically determine the minimum total degree threshold for forcing the square of a Hamilton cycle in a digraph. We also give a conjecture on the corresponding threshold for $k$th powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the $k$th power of a Hamilton cycle; although this minimum semi-degree condition is not tight, it does provide the correct order of magnitude of the threshold. Turán-type problems for oriented graphs are also discussed.

For $\ell \geq 3$, an $\ell$-uniform hypergraph is disperse if the number of edges induced by any set of $\ell +1$ vertices is 0, 1, $\ell$, or $\ell +1$. We show that every disperse $\ell$-uniform hypergraph on $n$ vertices contains a clique or independent set of size $n^{\Omega _{\ell }(1)}$, answering a question of the first author and Tomon. To this end, we prove several structural properties of disperse hypergraphs.

We describe the asymptotic behaviour of large degrees in random hyperbolic graphs for all values of the curvature parameter $\alpha$. We prove that, with high probability, the node degrees satisfy the following ordering property: the ranking of the nodes by decreasing degree coincides with the ranking of the nodes by increasing distance to the centre, at least up to any constant rank. In the sparse regime $\alpha>\tfrac{1}{2}$, the rank at which these two rankings cease to coincide is $n^{1/(1+8\alpha)+o(1)}$. We also provide a quantitative description of the large degrees by proving the convergence in distribution of the normalised degree process towards a Poisson point process. In particular, this establishes the convergence in distribution of the normalised maximum degree of the graph. A transition occurs at $\alpha = \tfrac{1}{2}$, which corresponds to the connectivity threshold of the model. For $\alpha < \tfrac{1}{2}$, the maximum degree is of order $n - O(n^{\alpha + 1/2})$, whereas for $\alpha \geq \tfrac{1}{2}$, the maximum degree is of order $n^{1/(2\alpha)}$. In the $\alpha < \tfrac{1}{2}$ and $\alpha > \tfrac{1}{2}$ cases, the limit distribution of the maximum degree belongs to the class of extreme value distributions (Weibull for $\alpha < \tfrac{1}{2}$ and Fréchet for $\alpha > \tfrac{1}{2}$). This refines previous estimates on the maximum degree for $\alpha > \tfrac{1}{2}$ and extends the study of large degrees to the dense regime $\alpha \leq \tfrac{1}{2}$.

We study the hypersimplex under the action of the symmetric group $S_n$ by coordinate permutation. We prove that its equivariant volume, given by the evaluation of its equivariant $H^*$-series at $1$, is the permutation character of decorated ordered set partitions under the natural action of $S_n$. This verifies a conjecture of Stapledon for the hypersimplex. To prove this result, we give a formula for the coefficients of the $H^*$-polynomial. Additionally, for the $(2,n)$-hypersimplex, we use this formula to show that trivial character need not appear as a direct summand of a coefficient of the $H^*$-polynomial, which gives a family of counterexamples to a different conjecture of Stapledon.

In this paper, we will investigate the following inequality

\begin{equation*}(a_{n+1}a_{n+2} - a_{n}a_{n+3})^2 -(a_{n+1}^2 -a_{n}a_{n+2})(a_{n+2}^2 - a_{n}a_{n+4}) \gt 0,\end{equation*}

which arises from the iterated Laguerre operator on functions. We will prove the sequence $\{a_n\}$ of a unified form given by Griffin, Ono, Rolen and Zagier asymptotically satisfies this inequality while the Maclaurin coefficients of the functions in Laguerre-Pólya class have not to possess this inequality. We also prove the companion version of this inequality. As a consequence, we show the Maclaurin coefficients of the Riemann Ξ-function asymptotically satisfy this property. Moreover, we make this approach effective and give the exact thresholds for the positivity of this inequalityfor the partition function, the overpartition function and the smallest part function.