Searcharxiv⌕ Search

arXiv subjects

Xian'an Jin

Publications and source records attributed to Xian'an Jin.

At least 19 recordsLinked to original sources
arXiv2026

Subtree Counts and the Maximum Mean Subtree Order

For a finite simple graph $G$ of positive order, let $s_k(G)$ be the number of its $k$-vertex tree subgraphs. We prove that, among graphs of a fixed order $n$, the ratios $s_k(G)/s_k(K_n)$ form a nonincreasing sequence in $k$. It follows that the complete graph maximizes the mean subtree order: $μ(G)\leqμ(K_n)$, with equality if and only if $G$ is complete. This monotonicity theorem also establishes the $s_{n-1}$-to-$s_n$ ratio conjecture posed in recent work on extremal mean subtree order. The proof lifts a known coupling of rooted Cayley-tree shapes of consecutive orders to uniform labeled tree subgraphs of $K_n$ that are nested at every sample point. Event inclusion gives the count inequalities, and a double-sum identity gives the mean bound.

math.CO↗
arXiv2026

A Fano framework for binary delta-matroids

Dunshee and Ellingham recently showed that seven natural properties of a cellularly embedded graph form a Fano-plane framework. We establish an analogous framework for binary delta-matroids. For a binary delta-matroid $D$ and $τ$, let $Z_3(D,τ)$ denote its associated binary tight $3$-matroid. The six outer points are represented by evenness or bipartiteness of $D$ and its global vertex-flip transforms. For the seventh point, we call $D$ $Z_3$-bipartite when every circuit of $Z_3(D,τ)$ has even cardinality. We show that the satisfied properties are precisely the nonzero vectors of a subspace of $\Ftwo^3$. For ribbon-graphic delta-matroids, $Z_3$-bipartiteness is equivalent to bipartiteness of the medial graph, so the construction recovers the Fano-plane framework for embedded graphs.

math.CO↗
arXiv2026

Two conjectures on graphs and their edge-path matrices

The edge-path matrix is a square matrix where each off-diagonal entry records the maximum number of edge-disjoint paths between the corresponding pair of vertices. Akbari et al. [On edge-path eigenvalues of graphs, Linear Multilinear Algebra 70 (2022) 2998-3008] proposed two conjectures: Conjecture 1 relates the edge-path matrix to an upper bound on the number of edges in the graph, while Conjecture 2 asserts that a graph is Eulerian if and only if all entries of its edge-path matrix are even. In this paper, we prove the two conjectures.

math.CO↗
arXiv2026

Proof of a Brown-Mol conjecture on subtree roots

The subtree polynomial of a tree is the generating function that enumerates its subtrees according to their orders. Brown and Mol conjectured that every subtree root of a tree of order $n\ge 2$ lies in the disk \[ \left\{z\in\mathbb C: |z|\le 1+\sqrt[n-1]{n-1} \right\}. \] We prove this conjecture by introducing a recursive comparison method based on an extremal problem over integer compositions. We further characterize the equality case: the upper bound is attained if and only if $n$ is even and the tree is the star; in this case the unique boundary root is $-1-\sqrt[n-1]{n-1}$. We also show that every nonzero subtree root $z$ satisfies \[ |z|>\sqrt[n-1]{n-1}-1. \] The lower bound is asymptotically sharp as $n\to\infty$. For odd $n$, although the upper bound is not attained, it is asymptotically sharp as $n\to\infty$.

math.CO↗
arXiv2026

Irreducibility of interlace polynomials

The factorisation of graph polynomials often reflects combinatorial decomposition. For a nonempty loopless graph $G$, we first prove that the two-variable interlace polynomial $q(G;x,y)$, introduced by Arratia, Bollobás and Sorkin, is irreducible over $\mathbb{C}[x,y]$ if and only if $G$ is connected, exactly paralleling the classical irreducibility theorem for the Tutte polynomial. The loopless hypothesis is essential: we construct an infinite family of connected looped graphs whose two-variable interlace polynomials are reducible. For a nonempty graph $G$, we prove that Courcelle's multivariate interlace polynomial $C_G(u,v;\mathbf{x},\mathbf{y})$ is irreducible over $\mathbb{C}[u,v,x_a,y_a:a\in V(G)]$ if and only if $G$ is connected.

math.CO↗
arXiv2026

A deletion-contraction formula and monotonicity properties for the polymatroid Tutte polynomial

The Tutte polynomial is a fundamental invariant of matroids. The polymatroid Tutte polynomial $\mathscr{T}_{P}(x,y)$, introduced by Bernardi, Kálmán, and Postnikov, is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this paper, we first obtain a deletion-contraction formula for $\mathscr{T}_{P}(x,y)$. Then we prove two natural properties of coefficientwise monotonicity, one for containment and one for minors, both for the interior polynomial $x^{n}\mathscr{T}_{P}(x^{-1},1)$ and the exterior polynomial $y^{n}\mathscr{T}_{P}(1,y^{-1})$, where $P$ is a polymatroid over $[n]$. We show by an example that these monotonicity properties do not extend to $\mathscr{T}_{P}(x,y)$. Using deletion-contraction, we obtain formulas for the coefficients of terms of degree $n-1$ in $\mathscr{T}_{P}(x,y)$. Finally, we characterize hypergraphs $\mathcal{H}=(V,E)$ such that the coefficient of $y^{k}$ in the exterior polynomial of the associated polymatroid $P_{\mathcal{H}}$ attains its maximal value $\binom{|V|+k-2}{k}$ for all $k$ up to some bound.

math.CO↗
arXiv2026

A direct proof of well-definedness for the polymatroid Tutte polynomial

For a polymatroid $P$ over $[n]$, Bernardi, Kálmán and Postnikov [\emph{Adv. Math.} 402 (2022) 108355] introduced the polymatroid Tutte polynomial $\mathscr{T}_{P}$ relying on the order $1<2<\cdots<n$ of $[n]$, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that $\mathscr{T}_{P}$ is equivalent to another polynomial that only depends on $P$. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.

math.CO↗
arXiv2026

Connectivity for slice-projections of connected polymatroids

It is well-known that deleting or contracting any element of a connected matroid always yields at least one connected minor. However, for a connected polymatroid, only two such elements can be guaranteed, proved by Hall in 2013. This note investigates the connectivity properties of slice-projections of connected polymatroids, which includes deletion and contraction. We establish that for any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected. We also obtain that the $j$-th slice-projection from the top of a polymatroid and $j$-th slice-projection from the bottom of its dual have the same connectedness. These results both extend existing connectedness theorems for graphs and matroids.

math.CO↗
arXiv2026

Partial-twuality polynomial interpolation for binary delta-matroids

Gross, Mansour and Tucker introduced the partial-twuality polynomials for ribbon graphs and investigated the interpolation property of these polynomials. The ribbon group generated by $δ$ and $τ$ acts on set systems as twist $\ast$ and loop complementation $\times$, yielding five nontrivial twuality operators: $ \{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast \}.$ Yan and Jin extended partial-twuality polynomials to set systems, yielding partial-$\bullet$ polynomials with $\bullet\in\{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast\}.$ For partial-$\ast$ polynomials, Zhao and Yan proved that this polynomial is either even, odd, or both even-interpolating and odd-interpolating for every binary delta-matroid. In this paper, we extend this interpolation property to all the remaining nontrivial partial-twualities of binary delta-matroids. Consequently, for every binary delta-matroid and every $\bullet\in\{\ast,\times,\ast \times ,\times\ast ,\ast \times \ast \}$, the partial-$\bullet$ polynomial is either even, odd, or both even-interpolating and odd-interpolating. We also provide examples to show that the binary assumption is essential.

math.CO↗
arXiv2026

Coefficients of univariate Tutte polynomials with one variable fixed

It is well known that the 2-variable Tutte polynomial of a graph $G$ includes chromatic polynomial and flow polynomial of $G$, i.e. the cases of $y=0$ and $x=0$. In 2013, Kálmán introduced the interior and exterior polynomials which generalized the cases of $y=1$ and $x=1$ of Tutte polynomials of graphs to hypergraphs, and further polymatroids. There have been some results on coefficients of these polynomials, which motivate us to study uniformly the coefficients of $T_M(x,t)$ and $T_M(t,y)$, where $T_M(x,y)$ denotes the Tutte polynomial of a matroid $M$ and $t$ is a fixed real number. In this paper, we introduce two mutually dual parameters $f_k(M)$ and $g_k(M)$ ($g_1(M)$ is the girth of $M$) for any nonnegative integer $k$, and obtain the following results: (1) Formulas for coefficients of the higher-degree terms (related to $g_2(M)$ and $f_2(M)$, respectively) of $T_M(x,t)$ and $T_M(t,y)$ in terms of circuits and hyperplanes of $M$; (2) when $0\leq t \leq 1$, coefficients of the more higher-degree terms (related to $g_1(M)$ and $f_1(M)$, respectively) of $T_M(x,t)$ and $T_M(t,y)$ are further simplified and characterized; (3) As applications, some known results in the cases $t=0$ and $t=1$ are derived and generalized, and the unimodality of these coefficients in (1) are proved when $t\leq 1$.

math.CO↗
arXiv2026

Combinatorial interpretations of Tutte polynomials at the point $(2,-1)$

Let $G$ be a simple connected graph, and let $T_{G}(x,y)$ be the Tutte polynomial of $G$. Motivated by the works in \cite{Ma}, we, in this paper, introduce the even-left spanning forests of $G$ and odd $G$-partitionable permutations, and show that $T_{G}(2,-1)$ is equal to both the number of even-left spanning forests of $G$ and the number of odd $G$-partitionable permutations. In particular, for a complete graph $K_n$, we prove that $T_{K_{n}}(2,-1)$ is the number of alternating permutations on $\{1,2,\dots,n+1\}$, using two distinct techniques: a recurrence relation and an explicit bijection construction.

math.CO↗
arXiv2026

Partial-twuality polynomials of matrices

The study of partial-twuality polynomials originates from the classical operations of geometric duality and Petrie duality on cellularly embedded graphs. These involutions generate the symmetric group $S_3$, and applying them to subsets of edges yields the notions of partial-(geometric) duality, partial-Petriality, and more generally, partial-twuality. In this paper, we generalize this theory of partial-twuality polynomials within the framework of matrix algebra. The key observation that the Euler genus of a bouquet under a partial-twuality can be expressed as a rank function of its adjacency matrix motivates and leads to the definition of a partial-twuality polynomial for an arbitrary square matrix over any field, thereby providing a universal algebraic counterpart to the topological polynomials. We then investigate basic properties of these polynomials, including product formulas, recursion relations, degrees, interpolation behaviors, and invariance and duality theorems under the matrix operations of pivoting and inversion. We conclude by posing some problems for further research.

math.CO↗
arXiv2026

On the maximum twist width of delta-matroids

For a ribbon graph $G$, let $γ(G)$ denote its Euler genus. Recently, Chen, Gross and Tucker [J. Algebraic Combin. 63 (2026) 13] derived a formula for the maximum partial-dual Euler-genus $\partialγ_M(G)$ of a ribbon graph $G$. Their key finding is that $\partialγ_M(G)$ can be achieved by a partial dual with respect to the edge set of a spanning quasi-tree. Moreover, they proposed the following problem: Given a ribbon graph $G$, is there a sequence of edges $e_1,e_2,\dots, e_k$ such that $γ(G^{\{e_1, e_2,\dots, e_k\}})=\partialγ_M(G)$ and such that the sequence $$γ(G), γ(G^{\{e_1\}}), \dots, γ(G^ {\{e_1, e_2,\dots, e_k\}})$$ rises monotonically (i.e., never decreasing) to $\partialγ_M(G)$? Delta-matroids are set systems that satisfy the symmetric exchange axiom and serve as a matroidal abstraction of ribbon graphs. In this paper, we first show that the maximum twist width of a set system can be attained by twisting one of its feasible sets, which extends the result of Chen, Gross and Tucker to set systems. Then we solve the delta-matroid version of their problem, thereby providing an affirmative answer to the original problem for ribbon graphs.

math.CO↗
arXiv2025

Matrix Quasi-tree Theorem

Building on prior work that established Matrix Quasi-tree Theorems for special embedded graphs, in this paper, we develop a comprehensive theory applicable to all embedded graphs. We introduce symbolic skew-adjacency matrices and reduction maps as key innovations, and prove that a specific polynomial derived from these matrices encodes all spanning quasi-trees of a bouquet. This result provides a complete analogue of the Matrix Tree Theorem for topological graph theory, with applications to quasi-tree enumeration in both orientable and non-orientable embedded graphs.

math.CO↗
arXiv2025

On the enumeration of connected sets in finite cylindrical lattice graphs

A connected set in a graph is a non-empty set of vertices that induces a connected subgraph. In an infinite lattice, a connected set is often referred to as a lattice animal, whose enumeration up to isomorphism is a classical problem in both combinatorics and statistical physics. In this paper, we focus on the enumeration of connected sets in finite lattice graphs, providing a link between combinatorial counting and structural connectivity in the system. For any positive integers $m,n$, let $N(P_m\times P_n)$ and $N(C_m\times P_n)$ denote the number of all connected sets in the $(m\times n)$-lattice graph $P_m\times P_n$ and $(m\times n)$-cylindrical lattice graph $C_m\times P_n $, respectively. In 2020, Vince derived enumeration formulas for $N(P_m\times P_2)$ and $N(C_m\times P_2)$, and highlighted the increasing difficulty of extending these calculation results to larger (cylindrical) lattice graphs. Recently, the authors of this paper have developed a method based on multi-step recurrence formulas to obtain the enumeration formula for $N(P_m\times P_n)$ with $m\le 4$. In this article, we apply a similar approach to derive the enumeration formula for $N(C_m\times P_n)$ with $m\le 7$. Further, for the general case, we establish an explicit and tight lower bound on the number of connected sets in the Cartesian product graph $G\times P_n $ for any connected graph $G$, by employing the transfer matrix method on a subclass of connected sets. Based on this, we perform an asymptotic analysis on several lattice graphs and show that $O(N(P_3\times P_n))=1.6694^{3n}$, $O(N(C_4\times P_n))=1.8014^{4n}$, and $O(N(C_5\times P_n))=1.7877^{5n}$.

math.CO↗
arXiv2025

Proof of a conjecture of Fomichev and Karev

We prove a conjecture of Fomichev and Karev [{European J. Combin.} 127 (2025) 104160] by showing the equality of two graph invariants: $φ$, defined via graph colorings, and $ψ$, derived from the $\mathfrak{sl}(2)$-weight system of its 2-dimensional irreducible representation.

math.CO↗
arXiv2025

Orbits and self-twuality in set systems and delta-matroids

We introduce a new group action on set systems, constructed as a semidirect product of a permutation group and a group generated by twist and loop complementation operations on a single element. This action extends the ribbon group framework of Abrams and Ellis-Monaghan from ribbon graphs to set systems, facilitating a systematic investigation of self-twuality. We prove that different forms of self-twuality propagate through orbits under the group action and establish a characterization of the orbit of a vf-safe delta-matroid via multimatroids. As an application, we analyze orbits of ribbon-graphic delta-matroids. Our work answers a question posed by Abrams and Ellis-Monaghan and provides a unified algebraic framework for studying self-twuality in combinatorial structures.

math.CO↗
arXiv2025

A recursive definition for the polymatroid Tutte polynomial

The Tutte polynomial is a significant invariant of graphs and matroids. It is well-known that it has three equivalent definitions: bases expansion, rank generating function, and deletion-contraction formula. The polymatroid Tutte polynomial $\mathscr{T}_{P}$ generalizes the Tutte polynomial from matroids to polymatroids $P$. In \emph{[Adv. Math. 402 (2022) 108355.]} and \emph{[J. Combin. Theory Ser. A 188 (2022) 105584]}, the authors provided bases expansion and rank generating function constructions for $\mathscr{T}_{P}$, respectively. In \emph{[Int. Math. Res. Not. 19 (2025) rnaf302]}, a recursive formula for $\mathscr{T}_{P}$ was obtained. In this paper, we show that the recursive formula itself can be used to define the polymatroid Tutte polynomial independently.

math.CO↗