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Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph~$G$ as a parameter. Let pe$(G)$ denote the value of ecc$_G(P)$ for a central path $P$ of $G$. We obtain tight upper bounds for pe$(G)$ in some graph classes. We show that pe$(G) \leq 1$ on biconvex graphs and that pe$(G) \leq 2$ on bipartite convex graphs. Moreover, we design algorithms that find such a path in linear time. On the other hand, by investigating the longest paths of a graph, we obtain tight upper bounds for pe$(G)$ on general graphs and $k$-connected graphs. Finally, we study the relation between a central path and a longest path in a graph. We show that on trees, and bipartite permutation graphs, a longest path is also a central path. Furthermore, for superclasses of these graphs, we exhibit counterexamples for this property.

\emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a \emph{non-additive} setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices $\mathcal{L}$ has a proto-exact structure, and furthermore that the subcategory of $\mathcal{L}$ consisting of finite lattices is equivalent to the category of finite $\mathbb{B}$-modules as proto-exact categories, where $\mathbb{B}$ is the \emph{Boolean semifield}. We also discuss some relations between $\mathcal{L}$ and geometric lattices (simple matroids) from this perspective. In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the \emph{Krasner hyperfield} $\mathbb{K}$, a well-known relation between finite $\mathbb{K}$-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.

For unweighted graphs, finding isometric embeddings is closely related to decompositions of $G$ into Cartesian products of smaller graphs. When $G$ is isomorphic to a Cartesian graph product, we call the factors of this product a factorization of $G$. When $G$ is isomorphic to an isometric subgraph of a Cartesian graph product, we call those factors a pseudofactorization of $G$. Prior work has shown that an unweighted graph's pseudofactorization can be used to generate a canonical isometric embedding into a product of the smallest possible pseudofactors. However, for arbitrary weighted graphs, which represent a richer variety of metric spaces, methods for finding isometric embeddings or determining their existence remain elusive, and indeed pseudofactorization and factorization have not previously been extended to this context. In this work, we address the problem of finding the factorization and pseudofactorization of a weighted graph $G$, where $G$ satisfies the property that every edge constitutes a shortest path between its endpoints. We term such graphs minimal graphs, noting that every graph can be made minimal by removing edges not affecting its path metric. We generalize pseudofactorization and factorization to minimal graphs and develop new proof techniques that extend the previously proposed algorithms due to Graham and Winkler [Graham and Winkler, '85] and Feder [Feder, '92] for pseudofactorization and factorization of unweighted graphs. We show that any $m$-edge, $n$-vertex graph with positive integer edge weights can be factored in $O(m^2)$ time, plus the time to find all pairs shortest paths (APSP) distances in a weighted graph, resulting in an overall running time of $O(m^2+n^2\log\log n)$ time. We also show that a pseudofactorization for such a graph can be computed in $O(mn)$ time, plus the time to solve APSP, resulting in an $O(mn+n^2\log\log n)$ running time., Comment: 37 pages, 10 figures

In this paper, we derive a recurrence relation of anti-forcing polynomial for catacondensed hexagonal systems.

In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. 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 $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.

A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order $n$ with size $m>\lfloor \frac{n^{2}}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for given order $n.$ Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size $m.$ In this paper, we prove $Q$-spectral versions of Mantel's theorem., 14 pages, 4 figures

A $k$-edge-weighting of $G$ is a mapping $\omega:E(G)\longrightarrow \{1,\ldots,k\}$. The edge-weighting of $G$ naturally induces a vertex-colouring $\sigma_{\omega}:V(G)\longrightarrow \mathbb{N}$ given by$\sigma_{\omega}(v)=\sum_{u\in N_G(v)}\omega(vu)$ for every $v\in V(G)$. The edge-weighting $\omega$ is neighbour sum distinguishing if it yields a proper vertex-colouring $\sigma_{\omega}$, \emph{i.e.}, $\sigma_{\omega}(u)\neq \sigma_{\omega}(v)$ for every edge $uv$ of $G$.We investigate a neighbour sum distinguishing edge-weighting with local constraints, namely, we assume that the set of edges incident to a vertex of large degree is not monochromatic. A graph is nice if it has no components isomorphic to $K_2$. We prove that every nice graph with maximum degree at most~5 admits a neighbour sum distinguishing $(\Delta(G)+2)$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights. Furthermore, we prove that every nice graph admits a neighbour sum distinguishing $7$-edge-weighting such that all the vertices of degree at least~6 are incident with at least two edges of different weights. Finally, we show that nice bipartite graphs admit a neighbour sum distinguishing $6$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights.

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$ for which $Q_q$ is positive semidefinite is fundamental in asymptotic spectral analysis of graphs from the viewpoint of quantum probability theory. Assume that $G$ has at least two vertices. Then $\pi(G)$ is easily seen to be a nonempty closed subset of the interval $[-1,1]$. In this note, we show that $\pi(G)=[-1,1]$ if and only if $G$ is isometrically embeddable into a hypercube (infinite-dimensional if $G$ is infinite) if and only if $G$ is bipartite and does not possess certain five-vertex configurations, an example of which is an induced $K_{2,3}$., Comment: 6 pages

The independence number of a square matrix $A$, denoted by $\alpha(A)$, is the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$ the $n\times n$ matrix with all entries equal to one. Given any integer $n$, we prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies $$\alpha(\phi(X))= \alpha(X) {\quad\rm for~ all\quad}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$\phi(X)=H\circ(P^TXP)\quad { \rm for~ all\quad}X\in S_n^+,$$ where $H=\phi(J_n-I_n)$ with all off-diagonal entries positive.

A graph $G$ is called interval colorable if it has a proper edge coloring with colors $1,2,3,\dots$ such that the colors of the edges incident to every vertex of $G$ form an interval of integers. Not all graphs are interval colorable; in fact, quite few families have been proved to admit interval colorings. In this paper we introduce and investigate a new notion, the interval coloring thickness of a graph $G$, denoted ${\theta_{\mathrm{int}}}(G)$, which is the minimum number of interval colorable edge-disjoint subgraphs of $G$ whose union is $G$. Our investigation is motivated by scheduling problems with compactness requirements, in particular, problems whose solution may consist of several schedules, but where each schedule must not contain any waiting periods or idle times for all involved parties. We first prove that every connected properly $3$-edge colorable graph with maximum degree $3$ is interval colorable, and using this result, we deduce an upper bound on ${\theta_{\mathrm{int}}}(G)$ for general graphs $G$. We demonstrate that this upper bound can be improved in the case when $G$ is bipartite, planar or complete multipartite and consider some applications in timetabling.

We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several lower bounds on $\mathcal{A}(G)$ and prove that these conjectures are true for specific classes of graphs such as triangulated graphs and graphs with maximum degree at most 2., 20 pages, 2 figures

We present some necessary and/or sufficient conditions for the positivity problem of three-term recurrence sequences. As applications we show the positivity of diagonal Taylor coefficients of some rational functions in a unified approach. We also establish a criterion for the positivity and log-convexity of such sequences., 13 pages

We prove that a graph has an r-bounded subdivision of a wheel if and only if it does not have a graph-decomposition of locality r and width at most two.

A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. A hole is called {\em even} if it has an even number of vertices. An {\em even-hole-free} graph is a graph with no even holes. A vertex of a graph is {\em bisimplicial} if the set of its neighbours is the union of two cliques. In an earlier paper \cite{bisimplicial}, Addario-Berry, Havet and Reed, with the authors, claimed to prove a conjecture of Reed, that every even-hole-free graph has a bisimplicial vertex, but we have recently been shown that the "proof" has a serious error. Here we give a proof using a different method.

Let $\mathcal{G}$ be a minor-closed graph class. We say that a graph $G$ is a $k$-apex of $\mathcal{G}$ if $G$ contains a set $S$ of at most $k$ vertices such that $G\setminus S$ belongs to $\mathcal{G}.$ We denote by $\mathcal{A}_k (\mathcal{G})$ the set of all graphs that are $k$-apices of $\mathcal{G}.$ We prove that every graph in the obstruction set of $\mathcal{A}_k (\mathcal{G}),$ i.e., the minor-minimal set of graphs not belonging to $\mathcal{A}_k (\mathcal{G}),$ has size at most $2^{2^{2^{2^{\mathsf{poly}(k)}}}},$ where $\mathsf{poly}$ is a polynomial function whose degree depends on the size of the minor-obstructions of $\mathcal{G}.$ This bound drops to $2^{2^{\mathsf{poly}(k)}}$ when $\mathcal{G}$ excludes some apex graph as a minor., Comment: 48 pages and 12 figures. arXiv admin note: text overlap with arXiv:2004.12692

For integers $k,t \geq 2$, and $1\leq r \leq t$ let $D_k^\times(r,t;n)$ be the number of parts among all $k$-indivisible partitions of $n$ (i.e., partitions where all parts are not divisible by $k$) of $n$ that are congruent to $r$ modulo $t$. Using Wright's circle method, we derive an asymptotic for $D_k^\times(r,t;n)$ as $n \to \infty$ when $k,t$ are coprime. The main term of this asymptotic does not depend on $r$, and so, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards different congruence classes modulo $t$. This induces an ordering on the congruence classes modulo $t$, which we call the $k$-indivisible ordering. We prove that for $k \geq \frac{6(t^2-1)}{\pi^2}$ the $k$-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when $k < \frac{6(t^2-1)}{\pi^2}$., Comment: 26 pages, 3 figures

The block reversal of a word $w$, denoted by $\mathtt{BR}(w)$, is a generalization of the concept of the reversal of a word, obtained by concatenating the blocks of the word in the reverse order. We characterize non-binary and binary words whose block reversal contains only rich words. We prove that for a binary word $w$, richness of all elements of $\mathtt{BR}(w)$ depends on $l(w)$, the length of the run sequence of $w$. We show that if all elements of $\mathtt{BR}(w)$ are rich, then $2\leq l(w)\leq 8$. We also provide the structure of such words.

The proper chromatic number $\Vec{\chi}(G)$ of a graph $G$ is the minimum $k$ such that there exists an orientation of the edges of $G$ with all vertex-outdegrees at most $k$ and such that for any adjacent vertices, the outdegrees are different. Two major conjectures about the proper chromatic number are resolved. First it is shown, that $\Vec{\chi}(G)$ of any planar graph $G$ is bounded (in fact, it is at most 14). Secondly, it is shown that for every graph, $\Vec{\chi}(G)$ is at most $O(\frac{r\log r}{\log\log r})+\tfrac{1}{2}\MAD(G)$, where $r=\chi(G)$ is the usual chromatic number of the graph, and $\MAD(G)$ is the maximum average degree taken over all subgraphs of $G$. Several other related results are derived. Our proofs are based on a novel notion of fractional orientations.

In 2017, Giudici, Li and the third author constructed the first known family of vertex-primitive $2$-arc-transitive digraphs of valency at least $2$. The smallest digraph in this family admits $\mathrm{PSL}_3(49)$ acting $2$-arc-transitively with vertex-stabilizer $\mathrm{A}_6$ and hence has $30758154560$ vertices. In this paper, we prove that this digraph is the vertex-primitive $2$-arc-transitive digraph of valency at least $2$ with fewest vertices.

Suppose $j_N(\tau)$ and $j_N^{*}(\tau)$ are the Hauptmoduln of the congruence subgroup $\Gamma_0(N)$ and the Fricke group $\Gamma^{*}_0(N)$, respectively. In [7], the authors predicted that, like Klein's $j$-function, the Fourier coefficients of $j_N(\tau)$ and $j_{N}^{*}(\tau)$ in some arithmetic progression are both even and odd with density $\frac{1}{2}$. In this article, we can find some arithmetic progression of $n$ where the Fourier coefficients of $j_6(\tau)$ (resp. $j_6^{*}(\tau)$ and $j_{10}(\tau)$) are almost always even. Furthermore, using Hecke eigenforms and Rogers-Ramanujan continued fraction, we obtain infinite families of congruences for $j_6(\tau)$, $j_6^{*}(\tau)$, $j_{10}(\tau),$ and $j_{10}^{*}(\tau)$., Comment: The comments are welcome!

A thoroughly studied problem in Extremal Graph Theory is to find the best possible density condition in a host graph $G$ for guaranteeing the presence of a particular subgraph $H$ in $G$. One such classical result, due to Bollob\'{a}s and Thomason, and independently Koml\'{o}s and Szemer\'{e}di, states that average degree $O(k^2)$ guarantees the existence of a $K_k$-subdivision. We study two directions extending this result. On the one hand, Verstra\"ete conjectured that the quadratic bound $O(k^2)$ would guarantee already two vertex-disjoint isomorphic copies of a $K_k$-subdivision. On the other hand, Thomassen conjectured that for each $k \in \mathbb{N}$ there is some $d = d(k)$ such that every graph with average degree at least $d$ contains a balanced subdivision of $K_k$, that is, a copy of $K_k$ where the edges are replaced by paths of equal length. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on $d(k)$ remains open. In this paper, we show that the quadratic bound $O(k^2)$ suffices to force a balanced $K_k$-subdivision. This gives the optimal bound on $d(k)$ needed in Thomassen's conjecture and implies the existence of $O(1)$ many vertex-disjoint isomorphic $K_k$-subdivisions, confirming Verstra\"ete's conjecture in a strong sense., Comment: 17 pages, 4 figures

A $0,1$ matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers $k$, there exists a square regular $0,1$ matrix with binary rank $k$, such that the Boolean rank of its complement is $k^{\widetilde{\Omega}(\log k)}$. Equivalently, the ones in the matrix can be partitioned into $k$ combinatorial rectangles, whereas the number of rectangles needed for any cover of its zeros is $k^{\widetilde{\Omega}(\log k)}$. This settles, in a strong form, a question of Pullman (Linear Algebra Appl., 1988) and a conjecture of Hefner, Henson, Lundgren, and Maybee (Congr. Numer., 1990). The result can be viewed as a regular analogue of a recent result of Balodis, Ben-David, G\"{o}\"{o}s, Jain, and Kothari (FOCS, 2021), motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers $k$, there exists a regular graph with biclique partition number $k$ and chromatic number $k^{\widetilde{\Omega}(\log k)}$., Comment: 21 pages

We study the classical expander codes, introduced by Sipser and Spielman [M. Sipser and D. A. Spielman, 1996]. Given any constants 0 < ��, �� < 1/2, and an arbitrary bipartite graph with N vertices on the left, M < N vertices on the right, and left degree D such that any left subset S of size at most �� N has at least (1-��)|S|D neighbors, we show that the corresponding linear code given by parity checks on the right has distance at least roughly {�� N}/{2 ��}. This is strictly better than the best known previous result of 2(1-��) �� N [Madhu Sudan, 2000; Viderman, 2013] whenever �� < 1/2, and improves the previous result significantly when �� is small. Furthermore, we show that this distance is tight in general, thus providing a complete characterization of the distance of general expander codes. Next, we provide several efficient decoding algorithms, which vastly improve previous results in terms of the fraction of errors corrected, whenever �� < 1/4. Finally, we also give a bound on the list-decoding radius of general expander codes, which beats the classical Johnson bound in certain situations (e.g., when the graph is almost regular and the code has a high rate). Our techniques exploit novel combinatorial properties of bipartite expander graphs. In particular, we establish a new size-expansion tradeoff, which may be of independent interests., LIPIcs, Vol. 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022), pages 43:1-43:3

Let $G$ be a graph of size $m$ and $\rho(G)$ be the spectral radius of its adjacency matrix. A graph is said to be $F$-free if it does not contain a subgraph isomorphic to $F$. In this paper, we prove that if $G$ is a $K_{2,r+1}$-free non-star graph with $m\geq (4r+2)^2+1$, then $\rho(G)\leq \rho(S_m^1)$, with equality if and only if $G\cong S_m^1$. Recently, Li, Sun and Wei showed that for any $\theta_{1,2,3}$-free graph of size $m\geq 8$, $\rho(G)\leq \frac{1+\sqrt{4m-3}}{2}$, with equality if and only if $G\cong S_{\frac{m+3}{2},2}$. However, this bound is not attainable when $m$ is even. We proved that if $G$ is $\theta_{1,2,3}$-free and $G\ncong S_{\frac{m+3}{2},2}$ with $m\geq 22$, then $\rho(G)\leq \rho(F_{m,1})$ if $m$ is even, with equality if and only if $G\cong F_{m,1}$, and $\rho(G)\leq \rho(F_{m,2})$ if $m$ is odd, with equality if and only if $G\cong F_{m,2}$., Comment: 15 pages, 3 figures

In this paper, we exhibit an irreducible Markov chain $M$ on the edge $k$-colorings of bipartite graphs based on certain properties of the solution space. We show that diameter of this Markov chain grows linearly with the number of edges in the graph. We also prove a polynomial upper bound on the inverse of acceptance ratio of the Metropolis-Hastings algorithm when the algorithm is applied on $M$ with the uniform distribution of all possible edge $k$-colorings of $G$. A special case of our results is the solution space of the possible completions of Latin rectangles., Comment: 20 pages, 2 figures

Here we introduce connectivity operators, namely, diffusion operators, general Laplacian operators, and general adjacency operators for hypergraphs. These operators are generalisations of some conventional notions of apparently different connectivity matrices associated with hypergraphs. In fact, we introduce here a unified framework for studying different variations of the connectivity operators associated with hypergraphs at the same time. Eigenvalues and corresponding eigenspaces of the general connectivity operators associated with some classes of hypergraphs are computed. Applications such as random walks on hypergraphs, dynamical networks, and disease transmission on hypergraphs are studied from the perspective of our newly introduced operators. We also derive spectral bounds for the weak connectivity number, degree of vertices, maximum cut, bipartition width, and isoperimetric constant of hypergraphs.

We improve Gilbert-Varshamov bound by graph spectral method. Gilbert graph $G_{q,n,d}$ is a graph with all vectors in $\mathbb{F}_q^n$ as vertices where two vertices are adjacent if their Hamming distance is less than $d$. In this paper, we calculate the eigenvalues and eigenvectors of $G_{q,n,d}$ using the properties of Cayley graph. The improved bound is associated with the minimum eigenvalue of the graph. Finally we give an algorithm to calculate the bound and linear codes which satisfy the bound.

Following Bradonji\'c and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the $2$-dimensional torus. In this model, the expected number of vertices of the graph is $n$, and the expected degree of a vertex is $a\log n$ for some fixed $a>1$. Each vertex is added with probability $p$ to a set $A_0$ of initially infected vertices. Vertices subsequently become infected if they have at least $ \theta a \log n $ infected neighbours. Here $p, \theta \in [0,1]$ are taken to be fixed constants. We show that if $\theta < (1+p)/2$, then a sufficiently large local outbreak leads with high probability to the infection spreading globally, with all but $o(n)$ vertices eventually becoming infected. On the other hand, for $ \theta > (1+p)/2$, even if one adversarially infects every vertex inside a ball of radius $O(\sqrt{\log n} )$, with high probability the infection will spread to only $o(n)$ vertices beyond those that were initially infected. In addition we give some bounds on the $(a, p, \theta)$ regions ensuring the emergence of large local outbreaks or the existence of islands of vertices that never become infected. We also give a complete picture of the (surprisingly complex) behaviour of the analogous $1$-dimensional bootstrap percolation model on the circle. Finally we raise a number of problems, and in particular make a conjecture on an `almost no percolation or almost full percolation' dichotomy which may be of independent interest., Comment: 38 pages, 8 figures, 9 pages appendices

For a hypergraph $\mathcal{H}$, define the minimum positive codegree $\delta_i^+(\mathcal{H})$ to be the largest integer $k$ such that every $i$-set which is contained in at least one edge of $\mathcal{H}$ is contained in at least $k$ edges. For $1\le s\le k,t$ and $t\le r$, we prove that for $n$-vertex $t$-intersecting $r$-graphs $\mathcal{H}$ with $\delta_{r-s}^+(\mathcal{H})>{k-1\choose s}$, the unique hypergraph with the maximum number of edges is the hypergraph $\mathcal{H}$ consisting of every edge which intersects a set of size $2k-2s+t$ in at least $k-s+t$ vertices provided $n$ is sufficiently large. This generalizes work of Balogh, Lemons, and Palmer who proved this for $s=t=1$, as well as the Erd\H{o}s-Ko-Rado theorem when $k=s$., Comment: 10 pages

We show that the first-order logical theory of the binary overlap-free words (and, more generally, the ${\alpha}$-free words for rational ${\alpha}$, $2 < {\alpha} \leq 7/3$), is decidable. As a consequence, many results previously obtained about this class through tedious case- based proofs can now be proved "automatically", using a decision procedure.

We introduce notions of Cheeger constants for graphons and graphings. We prove Cheeger and Buser inequalities for these. On the way we prove co-area formulae for graphons and graphings., Comment: 35 pgs, 4 figures; v2: 39 pgs, 5 figures. Relationships between Cheeger constant of finite weighted graph and associated canonical graphon added

We study a tropical linear regression problem consisting in finding the best approximation of a set of points by a tropical hyperplane. We establish a strong duality theorem, showing that the value of this problem coincides with the maximal radius of a Hilbert's ball included in a tropical polyhedron. We also show that this regression problem is polynomial-time equivalent to mean payoff games. We illustrate our results by solving an inverse problem from auction theory. In this setting, a tropical hyperplane represents the set of equilibrium prices. Tropical linear regression allows us to quantify the distance of a market to the set of equilibria, and infer secret preferences of a decision maker.

We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial., 34 pages, some corrections in the proofs and improvements in the exposition

Recently, Andrews, Bhattacharjee and Dastidar introduced the concept of [Formula: see text]-measure of an integer partition, and proved a surprising identity that the number of partitions of [Formula: see text] which have [Formula: see text]-measure [Formula: see text] is equal to the number of partitions of [Formula: see text] with a Durfee square of side [Formula: see text]. The authors asked for a bijective proof of this result and also suggested a further exploration of the properties of the number of partitions of [Formula: see text] which have [Formula: see text]-measure [Formula: see text] for [Formula: see text]. In this paper, we perform these tasks. That is, we obtain a short combinatorial proof of the result of Andrews, Bhattacharjee and Dastidar, and using this proof, we obtain a natural generalization for [Formula: see text]-measures.

We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k-1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$. The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and F��redi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k \ge 2^{n-d-1}$ we have $f(n,k,d)=2^d k - \left \lfloor \frac{k}{2^{n-d}} \right \rfloor$, while for $n > 2^{2^d k-k-d+1}$ we have $f(n,k,d)= n + 2^dk-d-2$, and also study the transition between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory., 15 pages

In this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in\mathbb{Z}$ and $x\equiv1\pmod 3$, then $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k} {2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{(-4)^k}\pmod{p^2},$$ and if $p\equiv1\pmod3$, then $$\sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{(-4)^k}\pmod{p^3},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ stands for the $n$th Franel number., 20 pages, revised some typos

In this paper we are interested in two kinds of (stacky) character varieties associated to a compact non-orientable surface. (A) We consider the quotient stack of the space of representations of the fundamental group of this surface to GL(n). (B) We choose a set of k-punctures on the surface and a generic k-tuple of semisimple conjugacy classes of GL(n), and we consider the stack of anti-invariant local systems on the orientation cover of the surface with local monodromies around the punctures given by the prescribed conjugacy classes. We compute the number of points of these spaces over finite fields from which we get a formula for their E-series (a certain specialization of the mixed Poincar\'e series). In case (B), we discuss the mixed Poincar\'e series when the surface is the real projective plane and k=1.

We answer to a question posed recently by B. Lovitz and F. Petrov, proving the conjectured sufficient minimality and uniqueness condition of the 3-tensor decomposition., Comment: 6 pages

We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatoric interpretation of the Kronecker coefficients. As an avenue to explore quantum supremacy and its implications for computational complexity theory, we outline experiments to detect non-vanishing Kronecker coefficients for hypothetical quantum realizations/simulations of these quantum systems. The correspondence between ribbon graphs and Belyi maps leads to an interpretation of these quantum mechanical systems in terms of quantum membrane world-volumes interpolating between string geometries., Version 1 - 49 pages; 12 pages of Appendices; 1 figure; revision V2 - added Lemma 1 which simplifies the proof of the main theorems; revision V3 - published version

We describe an approach, via Malle's permutation $\Psi$ on the set of irreducible characters $\text{Irr}(W)$, that gives a uniform derivation of the Chapuy-Stump formula for the enumeration of reflection factorizations of the Coxeter element. It also recovers its weighted generalization by delMas, Reiner, and Hameister, and further produces structural results for factorization formulas of arbitrary regular elements., Comment: 27 pages, comments very much welcome

We prove a recursive formula for plethysm coefficients of the form $a^\mu_{\lambda,(m)}$, generalising results on plethysms due to Bruns--Conca--Varbaro and de Boeck--Paget--Wildon. From this we deduce a stability result and resolve two conjectures of de Boeck concerning plethysms, as well as obtain new results on Sylow branching coefficients for symmetric groups for the prime 2. Further, letting $P_n$ denote a Sylow 2-subgroup of $S_n$, we show that almost all Sylow branching coefficients of $S_n$ corresponding to the trivial character of $P_n$ are positive., Comment: 35 pages, 6 figures; further explanation and detail added throughout, in particular in Section 5.1, results unchanged; typos corrected; acknowledgements updated

This paper is a follow-up to (arXiv:2203.03687), in which the first author studied primitive association schemes lying between a tensor power $\mathcal{T}_m^d$ of the trivial association scheme and the Hamming scheme $\mathcal{H}(m,d)$. A question which arose naturally in that study was whether all primitive fusions of $\mathcal{T}_m^d$ lie between $\mathcal{T}_{m^e}^{d/e}$ and $\mathcal{H}(m^d, d/e)$ for some $e \mid d$. This note answers this question positively provided that $m$ is large enough. We similarly classify primitive fusions of the $d$th tensor power of a Johnson scheme on $\binom{m}{k}$ points provided $m$ is large enough in terms of $k$ and $d$., 5 pages. Final version incorporating referees' comments. To appear in Algebraic Combinatorics

Let $n$ be a positive integer, and let $\rho_n = (n, n-1, n-2, \ldots, 1)$ be the ``staircase'' partition of size $N = {n+1 \choose 2}$. The Saxl conjecture asserts that every irreducible representation $S^\lambda$ of the symmetric group $S_N$ appears as a subrepresentation of the tensor square $S^{\rho_n} \otimes S^{\rho_n}$. In this short note we show that every irreducible representation of $S_N$ appears in the tensor cube $S^{\rho_n} \otimes S^{\rho_n} \otimes S^{\rho_n}$., Comment: 4 pages. Added an independent second proof of the main theorem

The classical Poincar\'e formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analogue of this formula. Along the way, we prove several foundational results about real tori with integral structures (and, therefore, tropical abelian varieties). For example, we prove a tropical version of the Appell-Humbert theorem. We also study various notions of equivalences between tropical cycles and their relation to one another.

In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between $0$ and $1$. Finally, we state several open conjectures that are motivated by empirical data.

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an \emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor problems in quasi-random $k$-graphs with minimum degree $\Omega(n^{k-1})$. In particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs $H(n)$ with minimum degree $\Omega(n^2)$ and minimum codegree $\Omega(n)$ but with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite $3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense quasi-random $3$-graphs of order $n$ with minimum codegree $\Omega(n)$ and $f\mid n$ have $F$-factors. That is, $1/8$ is the density threshold for ensuring all $3$-partite $3$-graphs $F$-factors in quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$. Moreover, we show that one can not replace the minimum codegree condition by a minimum vertex degree condition. In fact, we find that for any $p\in(0,1)$ and $n\ge n_0$, there exist $p$-dense quasi-random $3$-graphs of order $n$ with minimum degree $\Omega (n^2)$ having no $K_{2,2,2}$-factor. In particular, we study the optimal density threshold of $F$-factors for each $3$-partite $3$-graph $F$ in quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$., Comment: 22 pages. Accepted by Journal of Combinatorial Theory, Series B

Graph comparison is a certain type of condition on metric space encoded by a finite graph. We show that any nontrivial graph comparison implies one of Alexandrov's comparisons. The proof gives a complete description of graphs with trivial graph comparisons., Comment: English+Russian, 7+7 pages, 11+11 figures

A graph $H$ is an induced minor of a graph $G$ if $H$ can be obtained from $G$ by vertex deletions and edge contractions. We show that there is a function $f(k, d) = O(k^{10} + 2^{d^5})$ so that if a graph has treewidth at least $f(k, d)$ and maximum degree at most $d$, then it contains a $k \times k$-grid as an induced minor. This proves the conjecture of Aboulker, Adler, Kim, Sintiari, and Trotignon [Eur. J. Comb., 98, 2021] that any graph with large treewidth and bounded maximum degree contains a large wall or the line graph of a large wall as an induced subgraph. It also implies that for any fixed planar graph $H$, there is a subexponential time algorithm for maximum weight independent set on $H$-induced-minor-free graphs., 7 pages. To appear in JCTB

We characterize the simple connected graphs with the second largest eigenvalue less than 1/2, which consists of 13 classes of specific graphs. These 13 classes hint that $c_{2}\in [1/2, \sqrt{2+\sqrt{5}}]$, where $c_2$ is the minimum real number $c$ for which every real number greater than $c$ is a limit point in the set of the second largest eigenvalues of the simple connected graphs. We leave it as a problem., 36 pages, 2 tables

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph $G$ is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in $G$, and also in terms of subgraphs of $G$ that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of ``square-boundary'' amalgamations. These considerations also lead to an $\mathcal{O}(n\log n)$-time recognition algorithm to compute a decomposition of a planar median graph with $n$ vertices into cubes and square-graphs.