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Let $\lambda_2$ be the second largest eigenvalue of the adjacency matrix of a connected graph. In 2023, Li and Sun \cite{LiSun1} determined all the connected $\{K_{2,3}, K_4\}$-minor free graphs whose second largest eigenvalue $\lambda_2\le 1$. As a continuance of it, in this paper we completely identify all the connected $\{K_5,K_{3,3}\}$-minor free graphs without $C_3$ whose second largest eigenvalue does not exceed 1. This partially solves an open problem posed by Li and Sun \cite{LiSun1}: Characterize all connected planar graphs whose second largest eigenvalue is at most $1.$ Our main tools include the spectral theory and the local structure characterization of the planar graph with respect to its girth., Comment: 23 pages, 7 figures

We introduce a metric on the set of permutations of given order, which is a weighted generalization of Kendall's $\tau$ rank distance and study its properties. Using the edge graph of a permutohedron, we give a criterion which guarantees that a permutation lies metrically between another two fixed permutations. In addition, the conditions under which four points from the resulting metric space form a pseudolinear quadruple were found., Comment: 17 pages, 3 figures

This work shows a study about the structure of the cycles contained in a Minimal Strong Digraph (MSD). The structure of a given cycle is determined by the strongly connected components (or strong components, SCs) that appear after suppressing the arcs of the cycle. By this process and by the contraction of all SCs into single vertices we obtain a Hasse diagram from the MSD. Among other properties, we show that any SC conformed by more than one vertex (non trivial SC) has at least one linear vertex (a vertex with indegree and outdegree equal to 1) in the MSD (Theorem 1); that in the Hasse diagram at least one linear vertex exists for each non trivial maximal (resp. minimal) vertex (Theorem 2); that if an SC contains a number $\lambda$ of vertices of the cycle then it contains at least $\lambda$ linear vertices in the MSD (Theorem 3); and, finally, that given a cycle of length $q$ contained in the MSD, the number $\alpha$ of linear vertices contained in the MSD satisfies $\alpha \geq \lfloor (q+1)/2 \rfloor$ (Theorem 4).

An important theorem about the spectral Tur\'an problem of $K_{a,b}$ was largely developed in separate papers. Recently it was completely resolved by Zhai and Lin [J. Comb. Theory, Ser. B 157 (2022) 184-215], which also confirms a conjecture proposed by Tait [J. Comb. Theory, Ser. A 166 (2019) 42-58]. Here, the prior work is fully stated, and then generalized with a self-contained proof. The more complete result is then used to better understand the relationship between the $A_{\alpha}$-spectral radius and the structure of the corresponding extremal $K_{a,b}$-minor free graph., Comment: 31 pages, 2 figures

We consider polynomials of the form $\operatorname{h}_m(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]})$, where $\operatorname{h}_m$ is the complete homogeneous polynomial of degree $m$ and $y_j^{[\varkappa_j]}$ denotes $y_j$ repeated $\varkappa_j$ times. Using the decomposition of the generating function into partial fractions we represent such polynomials in the form \[ \operatorname{h}_m(y_1^{[\varkappa_1]},\ldots,y_n^{[\varkappa_n]}) =\sum_{j=1}^n \sum_{r=1}^{\varkappa_j} \binom{r+m-1}{r-1} A_{y,\varkappa,j,r} y_j^m, \] where $A_{y,\varkappa,j,r}$ are some coefficients that do not depend on $m$. We also provide an alternative proof using the inverse of the confluent Vandermonde matrix., Comment: 28 pages

Expository article on the problem of determining the maximum number of equiangular lines with a fixed angle, and the associated problem of second eigenvalue multiplicity in graphs., Comment: Invited feature article in the Notices of the American Mathematical Society

This paper concerns non-overlapping codes, block codes motivated by synchronisation and DNA-based storage applications. Most existing constructions of these codes do not account for the restrictions posed by the physical properties of communication channels. If undesired sequences are not avoided, the system using the encoding may start behaving incorrectly. Hence, we aim to characterise all non-overlapping codes satisfying two additional constraints. For the first constraint, where approximately half of the letters in each word are positive, we derive necessary and sufficient conditions for the code's non-expandability and improve known bounds on its maximum size. We also determine exact values for the maximum sizes of polarity-balanced non-overlapping codes having small block and alphabet sizes. For the other constraint, where long sequences of consecutive equal symbols lead to undesired behaviour, we derive bounds and constructions of constrained non-overlapping codes. Moreover, we provide constructions of non-overlapping codes that satisfy both constraints and analyse the sizes of the obtained codes.

We are interested in conjecturing the sign of the pizza quantity P(H,B(a,R)) for the irreducible Coxeter arrangements H of type A_n, where n=2,3 bmod 4, and type D_n, where n is odd. Our approach is to express the pizza quantity in terms of the pizza quantity of subarrangements known as 2-structures, and we obtain the first non-zero term in the multivariate Taylor expansion., Comment: 12 pages, 1 figure and 2 tables. To appear in Experimental Mathematics

In this paper, we investigate properties of a symmetric Toeplitz matrix and a Hankel matrix by studying the components of its graph. To this end, we introduce the notion of ``weighted Toeplitz graph" and ``weighted Hankel graph", which are weighted graphs whose adjacency matrix are a symmetric Toeplitz matrix and a Hankel matrix, respectively. By studying the components of a weighted Toeplitz graph, we show that the Frobenius normal form of a symmetric Toeplitz matrix is a direct sum of symmetric irreducible Toeplitz matrices. Similarly, by studying the components of a weighted Hankel matrix, we show that the Frobenius normal form of a Hankel matrix is a direct sum of irreducible Hankel matrices.

The metric representation of a vertex $u$ in a connected graph $G$ respect to an ordered vertex subset $W=\{\omega_1, \dots , \omega_n\}\subset V(G)$ is the vector of distances $r(u\vert W)=(d(u,\omega_1), \dots , d(u,\omega_n))$. A vertex subset $W$ is a resolving set of $G$ if $r(u\vert W)\neq r(v\vert W)$, for every $u,v\in V(G)$ with $u\neq v$. Thus, a resolving set with $n$ elements provides a set of metric representation vectors $S\subset \mathbb{Z}^n$ with cardinal equal to the order of the graph. In this paper, we address the reverse point of view, that is, we characterize the finite subsets $S\subset \mathbb{Z}^n$ that are realizable as the set of metric representation vectors of a graph $G$ with respect to some resolving set $W$. We also explore the role that the strong product of paths plays in this context. Moreover, in the case $n=2$, we characterize the sets $S\subset \mathbb{Z}^2$ that are uniquely realizable as the set of metric representation vectors of a graph $G$ with respect to a resolving set $W$.

We study a new extension formula for right Bol loops. We prove the necessary or sufficient conditions for the extension to be right Bol. We describe the most important invariants: right multiplication group, nuclei, and center. We show that the core is an involutory quandle which is the disjoint union of two isomorphic involutory quandles. We also derive further results on the structure group of the core of the extension., Comment: 17 pages

We show factorization formulas for a class of partition functions of rational six vertex model. First we show factorization formulas for partition functions under triangular boundary. Further, by combining the factorization formulas with the explicit forms of the generalized domain wall boundary partition functions by Belliard-Pimenta-Slavnov, we derive factorization formulas for partition functions under trapezoid boundary which can be viewed as a generalization of triangular boundary. We also discuss an application to emptiness formation probabilities under trapezoid boundary which admit determinant representations.

We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.

The computation of the domination-type parameters is a challenging problem in Cartesian product graphs. We present an algorithmic method to compute the $2$-domination number of the Cartesian product of a path with small order and any cycle, involving the $(\min,+)$ matrix product. We establish some theoretical results that provide the algorithms necessary to compute that parameter, and the main challenge to run such algorithms comes from the large size of the matrices used, which makes it necessary to improve the techniques to handle these objects. We analyze the performance of the algorithms on modern multicore CPUs and on GPUs and we show the advantages over the sequential implementation. The use of these platforms allows us to compute the $2$-domination number of cylinders such that their paths have at most $12$ vertices.

A vertex subset S of a graph G is said to 2-dominate the graph if each vertex not in S has at least two neighbors in it. As usual, the associated parameter is the minimum cardinal of a 2-dominating set, which is called the 2-domination number of the graph G. We present both lower and upper bounds of the 2-domination number of cylinders, which are the Cartesian products of a path and a cycle. These bounds allow us to compute the exact value of the 2-domination number of cylinders where the path is arbitrary, and the order of the cycle is n $\equiv$ 0(mod 3) and as large as desired. In the case of the lower bound, we adapt the technique of the wasted domination to this parameter and we use the so-called tropical matrix product to obtain the desired bound. Moreover, we provide a regular patterned construction of a minimum 2-dominating set in the cylinders having the mentioned cycle order., Comment: 19 pages, 4 figures

Understanding efficient modifications to improve network functionality is a fundamental problem of scientific and industrial interest. We study the response of network dynamics against link modifications on a weakly connected directed graph consisting of two strongly connected components: an undirected star and an undirected cycle. We assume that there are directed edges starting from the cycle and ending at the star (master-slave formalism). We modify the graph by adding directed edges of arbitrarily large weights starting from the star and ending at the cycle (opposite direction of the cutset). We provide criteria (based on the sizes of the star and cycle, the coupling structure, and the weights of cutset and modification edges) that determine how the modification affects the spectral gap of the Laplacian matrix. We apply our approach to understand the modifications that either enhance or hinder synchronization in networks of chaotic Lorenz systems as well as R\"ossler. Our results show that the hindrance of collective dynamics due to link additions is not atypical as previously anticipated by modification analysis and thus allows for better control of collective properties.

A hollow matrix is a square matrix whose diagonal entries are all equal to zero. Define $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$, where $\rho$ is the unique real root of $x^3 = x + 1$. We show that for every $\lambda < \lambda^*$, there exists $n \in \mathbb{N}$ such that if a symmetric hollow integer matrix has an eigenvalue less than $-\lambda$, then one of its principal submatrices of order at most $n$ does as well. However, the same conclusion does not hold for any $\lambda \ge \lambda^*$., Comment: 8 pages

The fork is the tree obtained from the claw $K_{1,3}$ by subdividing one of its edges once, and the antifork is its complement graph. We give a complete description of all graphs that do not contain the fork or antifork as induced subgraphs., Comment: This is an old paper. It was published in 2020 in Discrete Math where it was awarded Editors' choice

Let $\mathcal{K}$ be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $\mathbb{R}^n$, there is a well-studied notion of "ultrametric orthogonality" in $\mathcal{K}^n$. In this paper, motivated by a question of Erd{\H{o}}s in the real case, given integers $k \geq \ell \geq 2$, we investigate the maximum size of a subset $S \subseteq \mathcal{K}^n \setminus\{{\bf 0}\}$ satisfying the following property: for any $E \subseteq S$ of size $k$, there exists $F \subseteq E$ of size $\ell$ such that any two distinct vectors in $F$ are orthogonal. Other variants of this property are also studied.

We call a proper edge coloring of a graph $G$ a B-coloring if every 4-cycle of $G$ is colored with four different colors. Let $q_B(G)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on B-colorings, here we consider $q_B(G)$ for planar and outerplanar graphs in terms of the maximum degree $\Delta = \Delta(G)$. We prove that $q_B(G)\le 2\Delta+8$ for planar graphs, $q_B(G)\le 2\Delta$ for bipartite planar graphs and $q_B(G)\le \Delta+1$ for outerplanar graphs with $\Delta \ge 4$. We conjecture that, for $\Delta$ sufficiently large, $q_B(G)\le 2\Delta(G)$ for planar $G$ and $q_B(G)\le \Delta(G)$ for outerplanar $G$., Comment: 15 pages, 4 figures, to appear in Journal of Graph Theory

We consider extremal edge-coloring problems inspired by the theory of anti-Ramsey / rainbow coloring, and further by odd-colorings and conflict-free colorings. Let $G$ be a graph, and $F$ any given family of graphs. For every integer $n \geq |G|$, let $f(n,G|F)$ denote the smallest integer $k$ such that any edge coloring of the complete graph $K_n$ with at least $k$ colors forces a copy of $G$ in which each color class induces a member of $F$. Observe that in anti-Ramsey problems each color class is a single edge; i.e., $F=\{K_2\}$. In our previous paper [arXiv:2405.19812], attention was given mostly to the case where $F$ is hereditary under subgraph inclusion. In the present work we consider coloring problems inspired by odd-coloring and conflict-free coloring. As we shall see, dealing with these problems requires distinct additional tools to those used in our first paper on the subject. Among the many results introduced in this paper, we mention: (1) For every graph $G$, there exists a constant $c=c(G)$ such that in any edge coloring of $K_n$ with at least $cn$ colors there is a copy of $G$ in which every vertex $v$ is incident with an edge whose color appears only once among all edges incident with $v$. (2) In sharp contrast to the above result we prove that if $F$ is the class of all odd graphs (having vertices with odd degrees only) then $f(n,K_k|F)=(1+o(1))$ex$(n,K_{\lceil k/2 \rceil})$, which is quadratic for $k \geq 5$. (3) We exactly determine $f(n,G|F)$ for small graphs when $F$ belongs to several families representing various odd/even coloring constraints., Comment: 56 pages

In 1955, Greenwood and Gleason showed that the Ramsey number R(3, 3, 3) = 17 by constructing an edge-chromatic graph on 16 vertices in three colors with no triangles. Their technique employed finite fields. This same result was obtained later by using another technique. In this article, we examine the complete graph on 17 vertices, K17, which can be represented as a regular polygon of 17 sides with all its diagonals. We color each edge of K17 with one of the three colors, blue, red or yellow. The graph thus obtained is called complete trichromatic graph K17^(3) (the superscript determines the number of colors). A triangle contained in graph K17^(3) with edges colored with one and only one color is called monochromatic. It has been shown that for any coloring of the K17^(3) edges, K17^(3) contains at least one monochromatic triangle. This article examines the problem of determining the minimum number of monochromatic triangles with the same color contained in K17^(3)., Comment: 3 pages, 3 figures

We prove that $K_n+I$, the complete graph of an even order with a $1$-factor duplicated, admits a decomposition into $2$-factors, each a disjoint union of cycles of length $m \geq 5$ if and only if $m \mid n$, except possibly when $m$ is odd and $n=4m$. In addition, we show that $K_n+I$ admits a decomposition into $2$-factors, each a disjoint union of cycles of lengths $m_1, \ldots, m_t$, whenever $m_1, \ldots, m_t$ are all even.

We prove that if $G$ is a graph and $f(v) \leq 1/(d(v) + 1/2)$ for each $v\in V(G)$, then either $G$ has an independent set of size at least $\sum_{v\in V(G)}f(v)$ or $G$ contains a clique $K$ such that $\sum_{v\in K}f(v) > 1$. This result implies that for any $\sigma \leq 1/2$, if $G$ is a graph and every clique $K\subseteq V(G)$ has at most $(1 - \sigma)(|K| - \sigma)$ simplicial vertices, then $\alpha(G) \geq \sum_{v\in V(G)} 1 / (d(v) + 1 - \sigma)$. Letting $\sigma = 0$ implies the famous Caro-Wei Theorem, and letting $\sigma = 1/2$ implies that if fewer than half of the vertices in each clique of $G$ are simplicial, then $\alpha(G) \geq \sum_{v\in V(G)}1/(d(v) + 1/2)$, which is tight for the 5-cycle. When applied to the complement of a graph, this result implies the following new Tur\' an stability result. If $G$ is a $K_{r + 1}$-free graph with more than $(1 - 1/r)n^2/2 - n/4$ edges, then $G$ contains an independent set $I$ such that at least half of the vertices in $I$ are complete to $G - I$. Applying this stability result iteratively provides a new proof of the stability version of Tur\' an's Theorem in which $K_{r + 1}$-free graphs with close to the extremal number of edges are $r$-partite., Comment: 16 pages. Derived from the preprint arxiv:1811.11806v1

In 2023 we obtained a $Q$-polynomial structure for the projective geometry $L_N(q)$. In the present paper, we display a more general $Q$-polynomial structure for $L_N(q)$. Our new $Q$-polynomial structure is defined using a free parameter $\varphi$ that takes any positive real value. For $\varphi=1$ we recover the original $Q$-polynomial structure. We interpret the new $Q$-polynomial structure using the quantum group $U_{q^{1/2}}(\mathfrak{sl}_2)$ in the equitable presentation. We use the new $Q$-polynomial structure to obtain analogs of the four split decompositions that appear in the theory of $Q$-polynomial distance-regular graphs., Comment: 30 pages

In this paper, we propose a network whose nodes are labeled by the composite numbers and two nodes are connected by an undirected link if they are relatively prime to each other. As the size of the network increases, the network will be connected whenever the largest possible node index $n\geq 49$. To investigate how the nodes are connected, we analytically describe that the link density saturates to $6/\pi^2$, whereas the average degree increases linearly with slope $6/\pi^2$ with the size of the network. To investigate how the neighbors of the nodes are connected to each other, we find the shortest path length will be at most 3 for $49\leq n\leq 288$ and it is at most 2 for $n\geq 289$. We also derive an analytic expression for the local clustering coefficients of the nodes, which quantifies how close the neighbors of a node to form a triangle. We also provide an expression for the number of $r$-length labeled cycles, which indicates the existence of a cycle of length at most $O(\log n)$. Finally, we show that this graph sequence is actually a sequence of weakly pseudo-random graphs. We numerically verify our observed analytical results. As a possible application, we have observed less synchronizability (the ratio of the largest and smallest positive eigenvalue of the Laplacian matrix is high) as compared to Erd\H{o}s-R\'{e}nyi random network and Barab\'{a}si-Albert network. This unusual observation is consistent with the prolonged transient behaviors of ecological and predator-prey networks which can easily avoid the global synchronization., Comment: 23 pages, 7 figures

In this paper, we solve a classical counting problem for non-degenerate forms of symplectic and hermitian type defined on a vector space: given a subspace $\pi$, we find the number of non-singular subspaces that are trivially intersecting with $\pi$ and span a non-singular subspace with $\pi$. Lower bounds for the quantity of such pairs where $\pi$ is non-singular were first studied in ``Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)'', which was later improved in ``Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)'' and generalised in ``Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)''. In this paper, we derive explicit formulae, which allow us to give the exact proportion and improve the known lower bounds.

This article addresses a combinatorial problem with applications to algebraic geometry. To a convex lattice polytope $P$ and each of its integer dilations $kP$ one may associate the barycenter of its lattice points. This sequence of $k$-quantized barycenters converge to the (classical) barycenter of the polytope considered as a convex body. A basic question arises: is there a complete asymptotic expansion for this sequence? If so, what are its terms? This article initiates the study of this question. First, we establish the existence of such an expansion as well as determine the first two terms. Second, for Delzant lattice polytopes we use toric algebra to determine all terms using mixed volumes of virtual rooftop polytopes, or alternatively in terms of higher Donaldson--Futaki invariants. Third, for reflexive polytopes we show the quantized barycenters are colinear to first order, and actually colinear in the case of polygons. The proofs use Ehrhart theory, convexity arguments, and toric algebra. As applications we derive the complete asymptotic expansion of the Fujita--Odaka stability thresholds $\delta_k$ on arbitrary polarizations on (possibly singular) toric varieties. In fact, we show they are rational functions of $k$ for sufficiently large $k$. This gives the first general result on Tian's stabilization problem for $\delta_k$-invariants for (possibly singular) toric Fanos: $\delta_k$ stabilize in $k$ if and only if they are all equal to $1$, and when smooth if and only if asymptotically Chow semistable. We also relate the asymptotic expansions to higher Donaldson--Futaki invariants of test configurations motivated by Ehrhart theory, and unify in passing previous results of Donaldson, Ono, Futaki, and Rubinstein--Tian--Zhang., Comment: with an appendix by Yaxiong Liu. To appear in Math. Z

In this article we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.

This paper is a follow-up to the paper [Matrix periods and competition periods of Boolean Toeplitz matrices, {\it Linear Algebra Appl.} 672:228--250, (2023)]. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices $A=T_n\langle S; T \rangle$ satisfying the condition ($\star$) $\max S+\min T \le n$ and $\min S+\max T \le n$ are $d^+/d$ and $1$, respectively, where $d^+= \gcd (s+t \mid s \in S, t \in T)$ and $d = \gcd(d, \min S)$. In this paper, we claim that even if ($\star$) is relaxed to the existence of elements $s \in S$ and $t \in T$ satisfying $s+t \le n$ and $\gcd(s,t)=1$, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy ($\star$) but the relaxed condition. For example, for any positive integers $k, n$ with $2k+1 \le n$, it is easy to see that $T_n\langle k, n-k;k+1, n-k-1 \rangle$ does not satisfies ($\star$) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is $T_n\langle d^+,2d^+, \ldots, \lfloor n/d^+\rfloor d^+\rangle$.

The Friendship Paradox is a simple and powerful statement about node degrees in a graph (Feld 1991). However, it only applies to undirected graphs with no edge weights, and the only node characteristic it concerns is degree. Since many social networks are more complex than that, it is useful to generalize this phenomenon, if possible, and a number of papers have proposed different generalizations. Here, we unify these generalizations in a common framework, retaining the focus on undirected graphs and allowing for weighted edges and for numeric node attributes other than degree to be considered, since this extension allows for a clean characterization and links to the original concepts most naturally. While the original Friendship Paradox and the Weighted Friendship Paradox hold for all graphs, considering non-degree attributes actually makes the extensions fail around 50% of the time, given random attribute assignment. We provide simple correlation-based rules to see whether an attribute-based version of the paradox holds. In addition to theory, our simulation and data results show how all the concepts can be applied to synthetic and real networks. Where applicable, we draw connections to prior work to make this an accessible and comprehensive paper that lets one understand the math behind the Friendship Paradox and its basic extensions., Comment: 34 pages, 4 figures

We present two classes of level Eulerian posets. Both classes contain intervals of rank k+1 whose cd-index is the sum over all cd-monomials w of degree k and the coefficient of the monomial w is r to the power of the number of d's in w. We also show that the order complexes of every interval in the first class are homeomorphic to spheres., Comment: 18 pages, 2 figures

The Cayley sum graph $\Gamma_A$ of a set $A \subseteq \mathbb{Z}_n$ is defined to have vertex set $\mathbb{Z}_n$ and an edge between two distinct vertices $x, y \in \mathbb{Z}_n$ if $x + y \in A$. Green and Morris proved that if the set $A$ is a $p$-random subset of $\mathbb{Z}_n$ with $p = 1/2$, then the independence number of $\Gamma_A$ is asymptotically equal to $\alpha(G(n, 1/2))$ with high probability. Our main theorem is the first extension of their result to $p = o(1)$: we show that, with high probability, $$\alpha(\Gamma_A) = (1 + o(1)) \alpha(G(n, p))$$ as long as $p \ge (\log n)^{-1/80}$. One of the tools in our proof is a geometric-flavoured theorem that generalises Fre\u{i}man's lemma, the classical lower bound on the size of high dimensional sumsets. We also give a short proof of this result up to a constant factor; this version yields a much simpler proof of our main theorem at the expense of a worse constant., Comment: Small adjustments following a helpful referee report, version accepted for publication in the Journal of the London Mathematical Society

Erd\H{o}s and Graham found it conceivable that the best $n$-term Egyptian underapproximation of almost every positive number for sufficiently large $n$ gets constructed in a greedy manner, i.e., from the best $(n-1)$-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero. [This note solves Problem 206 on Bloom's website "Erd\H{o}s problems".], Comment: 7 pages; v2 corrects the history of the claim for rational numbers; v3 incorporates referee's suggestions. The author is grateful to Richard Green for discussing the paper and its background in the popular newsletter "A Piece of the Pi: mathematics explained"

A classical result of Erd\H{o}s and Rademacher (1955) indicates a supersaturation phenomenon. It says that if $G$ is a graph on $n$ vertices with at least $\lfloor {n^2}/{4} \rfloor +1$ edges, then $G$ contains at least $\lfloor {n}/{2}\rfloor$ triangles. We prove a spectral version of Erd\H{o}s--Rademacher's theorem. Moreover, Mubayi [Adv. Math. 225 (2010)] extends the result of Erd\H{o}s and Rademacher from a triangle to any color-critical graph. It is interesting to study the extension of Mubayi from a spectral perspective. However, it is not apparent to measure the increment on the spectral radius of a graph comparing to the traditional edge version (Mubayi's result). In this paper, we provide a way to measure the increment on the spectral radius of a graph and propose a spectral version on the counting problems for color-critical graphs., Comment: 27 pages, 5 figures. Any comments and suggestions are welcome

In this paper, the association scheme defined on the flags of a finite generalized quadrangle is considered. All possible fusions of this scheme are listed, and a full description for those of classes 2 and 3 is given. Furthermore, it is showed that an association scheme with appropriate parameters must arise from the flags of a generalized quadrangle. The same is done for one of its 4-class symmetric fusion.

In this paper we relate a well-known in symplectic geometry compactification of the space of symmetric bilinear forms considered as a chart of the Lagrangian Grassmannian to the specific compactifications of the space of electrical networks in the disc obtained in \cite{L}, \cite{CGS} and \cite{BGKT}. In particular, we state an explicit connection between these works and describe some of the combinatorics developed there in the language of symplectic geometry. We also show that the combinatorics of the concordance vectors forces the uniqueness of the symplectic form, such that corresponding points of the Grassmannian are isotropic. We define a notion of Lagrangian concordance which provides a construction of the compactification of the space of electrical networks in the positive part of the Lagrangian Grassmannian bypassing the construction from \cite{L}., Comment: Minor corrections

We give a closed form for $quotients$ of truncated basic hypergeometric series where the base $q$ is evaluated at roots of unity., Comment: $1+2+\dots+N = 1\cdot2\dotsb N$ pages

In the game of Cops and Robbers, one of the most useful results is that an isometric path in a graph can be guarded by one cop. In this paper, we introduce the concept of wide shadow in a subgraph, and use it to characterize all 1-guardable graphs. As an application, we show that 3 cops can capture a robber in any planar graph with the added restriction that at most two cops can move simultaneously, proving a conjecture of Yang and strengthening a classical result of Aigner and Fromme.

The use of graph centrality measures applied to biological networks, such as protein interaction networks, underpins much research into identifying key players within biological processes. This approach however is restricted to dyadic interactions and it is well-known that in many instances interactions are polyadic. In this study we illustrate the merit of using hypergraph centrality applied to a hypernetwork as an alternative. Specifically, we review and propose an extension to a recently introduced node and edge nonlinear hypergraph centrality model which provides mutually dependent node and edge centralities. A Saccharomyces Cerevisiae protein complex hypernetwork is used as an example application with nodes representing proteins and hyperedges representing protein complexes. The resulting rankings of the nodes and edges are considered to see if they provide insight into the essentiality of the proteins and complexes. We find that certain variations of the model predict essentiality more accurately and that the degree-based variation illustrates that the centrality-lethality rule extends to a hypergraph setting. In particular, through exploitation of the models flexibility, we identify small sets of proteins densely populated with essential proteins. One of the key advantages of applying this model to a protein complex hypernetwork is that it also provides a classification method for protein complexes, unlike previous approaches which are only concerned with classifying proteins., Comment: 20 pages, 10 figures

Given an integer partition $(h_1,h_2,\dots,h_k)$ of $n$, is it possible to find an order $n$ latin square with $k$ disjoint subsquares of orders $h_1,\dots,h_k$? This question was posed by L.Fuchs and is only partially solved. Existence has been determined in general when $k\leq 4$, and in this paper we will complete the case when $k=5$. We also prove some less general results for partitions with $k=5$., Comment: 15 pages, 17 figures

By constructing suitable nonnegative exponential sums we give upper bounds on the cardinality of any set $B_q$ in cyclic groups $\ZZ_q$ such that the difference set $B_q-B_q$ avoids cubic residues modulo $q$., Comment: 8 pages

The notion of $P$-stability of an infinite set of degree sequences plays influential role in approximating the permanents, rapidly sampling the realizations of graphic degree sequences, or even studying and improving network privacy. While there exist several known sufficient conditions for $P$-stability, we don't know any useful necessary condition for it. We also do not have good insight of possible structure of $P$-stable degree sequence families. At first we will show that every known infinite $P$-stable degree sequence set, described by inequalities of the parameters $n, c_1, c_2, \Sigma$ (the sequence length, the maximum and minimum degrees and the sum of the degrees) is ,,fully graphic" meaning that every degree sequence from the region with an even degree sum, is graphic. Furthermore, if $\Sigma$ does not occur in the determining inequality, then the notions of $P$-stability and full graphicality will be proved equivalent. In turns, this equality provides a strengthening of the well-known theorem of Jerrum, McKay and Sinclair about $P$-stability, describing the maximal $P$-stable sequence set by $n, c_1, c_2$. Furthermore we conjecture that similar equivalences occur in cases if $\Sigma$ also part of the defining inequality., Comment: 23 pages, 1 figure

It is shown that a normalized complex Hadamard matrix of order $6$ having three distinct columns, each containing at least one $-1$ entry necessarily belongs to the transposed Fourier family, or to the family of $2$-circulant complex Hadamard matrices. The proofs rely on solving polynomial system of equations by Gr\"obner basis techniques, and make use of a structure theorem concerning regular Hadamard matrices. As a consequence, members of these two families can be easily recognized in practice. In particular, one can identify complex Hadamard matrices appearing in known triplets of pairwise mutually unbiased bases in dimension $6$., Comment: 17 pages, preprint

In this paper, we introduce saturation and semisaturation functions of sequences, and we prove a number of fundamental results about these functions. Given a forbidden sequence $u$ with $r$ distinct letters, we say that a sequence $s$ on a given alphabet is $u$-saturated if $s$ is $r$-sparse, $u$-free, and adding any letter from the alphabet to an arbitrary position in $s$ violates $r$-sparsity or induces a copy of $u$. We say that $s$ is $u$-semisaturated if $s$ is $r$-sparse and adding any letter from the alphabet to $s$ violates $r$-sparsity or induces a new copy of $u$. Let the saturation function $\operatorname{Sat}(u, n)$ denote the minimum possible length of a $u$-saturated sequence on an alphabet of size $n$, and let the semisaturation function $\operatorname{Ssat}(u, n)$ denote the minimum possible length of a $u$-semisaturated sequence on an alphabet of size $n$. For alternating sequences, we determine both the saturation function and the semisaturation function up to a constant multiplicative factor. We show for every sequence that the semisaturation function is always either $O(1)$ or $\Theta(n)$. For the saturation function, we show that every sequence $u$ has either $\operatorname{Sat}(u, n) \ge n$ or $\operatorname{Sat}(u, n) = O(1)$. For every sequence with $2$ distinct letters, we show that the saturation function is always either $O(1)$ or $\Theta(n)$.

This paper explores Iwasawa theory from a graph theoretic perspective, focusing on the algebraic and combinatorial properties of Cayley graphs. Using representation theory, we analyze Iwasawa-theoretic invariants within $\mathbb{Z}_\ell$-towers of Cayley graphs, revealing connections between graph theory, number theory, and group theory. Key results include the factorization of associated Iwasawa polynomials and the decomposition of $\mu$- and $\lambda$-invariants. Additionally, we apply these insights to complete graphs, establishing conditions under which these invariants vanish., Comment: Version 2: 23 pages, minor corrections following referee reports. Example 2 added. Accepted for publication in Research in the Mathematical Sciences

The pattern $(k_1, k_2, \dots, k_\ell)$ is defined to have at least $k_1$ consecutive $1$'s followed by at least $k_2$ consecutive $2$'s, $\dots$, followed by at least $k_\ell$ consecutive $\ell$'s. By iteratively applying the method that was developed previously to decouple the combinatorial complexity involved in studying complicated patterns in random sequences, the distribution of pattern $(k_1, k_2, \dots, k_\ell)$ is derived for arbitrary $\ell$. Numerical examples are provided to illustrate the results.

We give an elementary account of the notion of Y-frieze patterns, explain some of their properties, and reveal their connection with Coxeter's frieze patterns.

In the classical probability model, let $f(n)$ be the maximum number of pairwise independent events for the sample space with $n$ sample points. The determination of $f(n)$ is equivalent to the problem of determining the maximum cardinality of specific intersecting families on the set $\{1,2,\ldots,n\}$ . We show that $f(n)\leq n+1$, and $f(n)=n+1$ if there exists a Hadamard matrix of order $n$.

In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of $GF(q)$-chordal matroids as those matroids that can be constructed from projective geometries over $GF(q)$ by a sequence of generalized parallel connections across projective geometries over $GF(q)$. Our main result showed that when $q=2$, such matroids have no induced minor in $\{M(C_4),M(K_4)\}$. In this paper, we show that the class of $GF(2)$-chordal matroids coincides with the class of binary matroids that have none of $M(K_4)$, $M^*(K_{3,3})$, or $M(C_n)$ for $n\geq 4$ as a flat. We also show that $GF(q)$-chordal matroids can be characterized by an analogous result to Rose's 1970 characterization of chordal graphs as those that have a perfect elimination ordering of vertices.