Daniele D'Angeli, Alfredo Donno
We compute the complexity of two infinite families of finite graphs: the Sierpiński graphs, which are finite approximations of the well-known Sierpińsky gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.
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A spanning tree T in a finite planar connected graph G determines a dual spanning tree T* in the dual graph G such that T and T* do not intersect. We show that it is not always possible to find T in G, such that the diameters of T and T* are both within a uniform multiplicative constant (independent of G) of the diameters of their ambient graphs.
The bipartite analogue of a classical spanning tree enumeration formula, Boolean functions, and their applications to counting odd spanning trees
本文通过经典生成树计数公式和布尔函数，给出了计算完全图中奇生成树的简单证明，并将其推广到完全二分图中。
The weighted spanning tree enumerator of a graph $G$ with weighted edges is the sum of the products of edge weights over all the spanning trees in $G$. In the special case that all of the edge weights equal $1$, the weighted spanning tree enumerator counts the number of spanning trees in $G$. The Weighted Matrix-Tree Theorem asserts that the weighted spanning tree enumerator can be calculated from the determinant of a reduced weighted Laplacian matrix of $G$. That determinant, however, is not always easy to compute. In this paper, we show how two well-known results from linear algebra, the Mat…
For any graph $G$ of order $n$, the spanning tree packing number \emph{$STP(G)$}, is the maximum number of edge-disjoint spanning trees contained in $G$. In this paper, we obtain some sharp lower bounds for the spanning tree numbers of Cartesian product graphs and Lexicographic product graphs.
On spanning trees whose degrees are congruent to one modulo $\ell$
本文研究度数模ℓ余1的生成树，通过构造性贪心算法证明满足特定条件的图存在此类生成树，简化了奇生成树的相关证明，并推导了完全图和完全二部图中该类生成树的数量公式。
New Ore-type conditions for Hamilton cycles and spanning trees with few leaves are presented.
We classify connected spanning convex subgraphs of the square cycles. We then show that every spanning tree of $C_n^2$ is contained in a unique nontrivial connected spanning convex subgraph of $C_n^2$. As a result, we obtain a purely combinatorial derivation of the formula for the number of spanning trees of the square cycles.
The main purpose of the paper is to develop an approach to evaluation or estimation of the spanning tree congestion of planar graphs. This approach is used to evaluate the spanning tree congestion of triangular grids.
Halin conjectured 20 years ago that a graph has a normal spanning tree if and only if every minor of it has countable colouring number. We prove Halin's conjecture. This implies a forbidden minor characterisation for the property of having a normal spanning tree.
Schmidt characterised the class of rayless graphs by an ordinal rank function, which makes it possible to prove statements about rayless graphs by transfinite induction. Halin asked whether Schmidt's rank function can be generalised to characterise other important classes of graphs. We answer Halin's question in the affirmative. Another largely open problem raised by Halin asks for a characterisation of the class of graphs with an end-faithful spanning tree. A well-studied subclass is formed by the graphs with a normal spanning tree. We determine a larger subclass, the class of normally tracea…
For integer $k\geq2,$ a spanning $k$-ended-tree is a spanning tree with at most $k$ leaves. Motivated by the closure theorem of Broersma and Tuinstra [Independence trees and Hamilton cycles, J. Graph Theory 29 (1998) 227--237], we provide tight spectral conditions to guarantee the existence of a spanning $k$-ended-tree in a connected graph of order $n$ with extremal graphs being characterized. Moreover, by adopting Kaneko's theorem [Spanning trees with constraints on the leaf degree, Discrete Appl. Math. 115 (2001) 73--76], we also present tight spectral conditions for the existence of a spann…
A tree is called k-ended tree if it has at most k leaves, where a leaf is a vertex of degree one. In this paper we prove that every 3-regular connected graph with n vertices such that n is greater than 8 has spanning sub tree with at most [(2n+4)/9]-ended tree. At the end we give a conjecture about spanning k-ended trees on 3-regular connected graphs.
In this paper we give a linear time algorithm for computing the number of spanninig trees in double nested graphs.
We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of edge-disjoint spanning trees in G. These graphs are of interest for the k-tree protocol of Itai and Rodeh [Inform. and Comput. 79 (1988), 43-59].
Spectral conditions for spanning $k$-trees or $k$-ended-trees of $t$-connected graphs
本文研究 $t$-连通图中生成 $k$-树与生成 $k$-端树的存在性，给出若干新的谱充分条件，并推广和改进了多项已有结果。
Maximum number of spanning trees and connectivity: Graphs with a fixed minimum degree and bipartite graphs
本文研究了具有固定最小度数的图和双图在最大生成树数目方面的特性，确定了具有特定连通性和阶数的图结构。
On enumeration of spanning trees of complete multipartite graphs containing a fixed spanning forest
本文提出了一种计算包含给定生成森林的完全多重图生成树数目的行列式公式，并推广了完全二分图的相关结果。
Let T1, T2,.... Tk be spanning trees in a graph G. If for any pair of vertices u and v of G, the paths between u and v in every Ti( 0 < i < k+1) do not contain common edges then T1, T2,.... Tk are called edge-disjoint spanning trees in G. The design of multiple edge-disjoint spanning trees has applications to the reliable communication protocols. The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube, possesses some properties superior to those of the hypercube. For AQn (n > 2), construction of n-1 edge-disjoint spanning trees is given the result is optimal with respect…
We give an Ore-Type condition sufficient for a graph G to have a spanning tree with a specified degree sequence.
This note derives the characteristic polynomial of a graph that represents nonjump moves in a generalized game of checkers. The number of spanning trees is also determined.
A spanning tree without a vertex of degree two is called a Hist which is an abbreviation for homeomorphically irreducible spanning tree. We provide a necessary condition for the existence of a Hist in a cubic graph. As one consequence, we answer affirmatively an open question on Hists by Albertson, Berman, Hutchinson and Thomassen.
We present an explicit connected spanning structure that appears in a random graph just above the connectivity threshold with high probability.
A 2-tree is a graph that can be formed by starting with a triangle and iterating the operation of making a new vertex adjacent to two adjacent vertices of the existing graph. Leizhen Cai asked in 1995 whether every maximal planar graph contains a spanning 2-tree. We answer this question in the negative by constructing an infinite class of maximal planar graphs that have no spanning 2-tree.
On stability of spanning tree degree enumerators
研究了连通图生成树度数枚举多项式的实稳定性，证明其充要条件为该图是距离遗传图。
We give an Ore-type condition sufficient for a graph G to have a spanning tree with small degrees and with few leaves.
Completely independent spanning trees in a graph $G$ are spanning trees of $G$ such that for any two distinct vertices of $G$, the paths between them in the spanning trees are pairwise edge-disjoint and internally vertex-disjoint. In this paper, we present a tight lower bound on the maximum number of completely independent spanning trees in $L(G)$, where $L(G)$ denotes the line graph of a graph $G$. Based on a new characterization of a graph with $k$ completely independent spanning trees, we also show that for any complete graph $K_n$ of order $n \geq 4$, there are $\lfloor \frac{n+1}{2} \rflo…
Albertson, Berman, Hutchinson, and Thomassen showed in 1990 that there exist highly connected graphs in which every spanning tree contains vertices of degree 2. Using a result of Alon and Wormald, we show that there exists a natural number $d$ such that every graph of minimum degree at least $d$ contains a spanning tree without adjacent vertices of degree 2. Moreover, we prove that every graph with minimum degree at least 3 has a spanning tree without three consecutive vertices of degree 2.
In the present paper we consider the problem of constructing all the projective rooted spanning trees of a given graph. We propose an algorithm based on reducing this problem to the problem of constructing all the maximal independent sets of a certain derived graph. We also offer an algorithm for step-by-step growing the spanning trees that sifts, at each step, any sub-trees non-extendable to projective spanning trees. The proposed algorithms were applied for analyzing the possible variants of the syntactic subordination of nouns through prepositions .
A tree with at most $k$ leaves is called a $k$-ended tree. A spanning 2-ended tree is a Hamilton path. A Hamilton cycle can be considered as a spanning 1-ended tree. The earliest result concerning spanning trees with few leaves states that if $k$ is a positive integer and $G$ is a connected graph of order $n$ with $d(x)+d(y)\ge n-k+1$ for each pair of nonadjacent vertices $x,y$, then $G$ has a spanning $k$-ended tree. In this paper, we improve this result in two ways, and an analogous result is proved for dominating $k$-ended trees based on the generalized parameter $t_k$ - the order of a larg…
We consider the number of spanning trees in circulant graphs of $βn$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $βn$ factors, while we derive a formula of $β-1$ factors. Using the same trick, we also derive a formula for the number of spanning trees in discrete tori. Moreover, the spanning tree entropy of circulant graphs with fixed and non-fixed generators is compared.
We present two short proofs for Diestel's criterion that a connected graph has a normal spanning tree provided it contains no subdivision of a countable clique in which every edge has been replaced by uncountably many parallel edges.
We prove a refinement of the tree packing theorem by Tutte/Nash-Williams for finite graphs. This result is used to obtain a similar result for end faithful spanning tree packings in certain infinite graphs and consequently to establish a sufficient Hamiltonicity condition for the line graphs of such graphs.
Spectral condition for spanning $k$-ended trees in $t$-connected graphs
该研究针对$t$-连通图中生成$k$-端树的存在性问题，给出了紧的谱半径条件，推广了已有相关研究结果。
In this paper we study the flow-property of graphs containing a spanning triangle-tree. Our main results provide a structure characterization of graphs with a spanning triangle-tree admitting a nowhere-zero $3$-flow. All these graphs without nowhere-zero $3$-flows are constructed from $K_4$ by a so-called bull-growing operation. This generalizes a result of Fan et al. in 2008 on triangularly-connected graphs and particularly shows that every $4$-edge-connected graph with a spanning triangle-tree has a nowhere-zero $3$-flow. A well-known classical theorem of Jaeger in 1979 shows that every grap…
Using the special value at $u=1$ of the Artin-Ihara $L$-function, we give a short proof of the count of the number of spanning trees in the $n$-cube.
Spanning \(k\)-trees and the colorful Carathéodory theorem
将 Blagojević 关于二分生成树与球楔的彩色 Carathéodory 定理推广到生成 \(k\)-树与球楔的并，并用初等方法证明。
We give a proof for sharp estimate for the number of spanning trees using linear algebra and generalize this bound to multigraphs. In addition, we show that this bound is tight for complete graphs. In addition, we give estimates for number of spanning trees in specific type of graphs such as product of graphs or Cartesian product of two graphs.
In this short note, we study pairwise edge-disjoint rainbow spanning trees in properly edge-coloured complete graphs, where a graph is rainbow if its edges have distinct colours. Brualdi and Hollingsworth conjectured that every $K_n$ properly edge-coloured by $n-1$ colours has $n/2$ edge-disjoint rainbow spanning trees. Kaneko, Kano and Suzuki later suggested this should hold for every properly edge-coloured $K_n$. Improving the previous best known bound, we show that every properly edge-coloured $K_n$ contains $Ω(n)$ pairwise edge-disjoint rainbow spanning trees. Independently, Pokrovskiy and…
A spanning tree of a properly edge-colored complete graph, $K_n$, is rainbow provided that each of its edges receives a distinct color. In 1996, Brualdi and Hollingsworth conjectured that if $K_{2m}$ is properly $(2m-1)$-edge-colored, then the edges of $K_{2m}$ can be partitioned into $m$ rainbow spanning trees except when $m=2$. By means of an explicit, constructive approach, in this paper we construct $\lfloor \sqrt{6m+9}/3 \rfloor$ mutually edge-disjoint rainbow spanning trees for any positive value of $m$. Not only are the rainbow trees produced, but also some structure of each rainbow spa…
We show that a graph $G$ has a normal spanning tree if and only if its vertex set is the union of countably many sets each separated from any subdivided infinite clique in $G$ by a finite set of vertices. This proves a conjecture by Brochet and Diestel from 1994, giving a common strengthening of two classical normal spanning tree criterions due to Jung and Halin. Moreover, our method gives a new, algorithmic proof of Halin's theorem that every connected graph not containing a subdivision of a countable clique has a normal spanning tree.
Random spanning trees of a graph $G$ are governed by a corresponding probability mass distribution (or "law"), $μ$, defined on the set of all spanning trees of $G$. This paper addresses the problem of choosing $μ$ in order to utilize the edges as "fairly" as possible. This turns out to be equivalent to minimizing, with respect to $μ$, the expected overlap of two independent random spanning trees sampled with law $μ$. In the process, we introduce the notion of homogeneous graphs. These are graphs for which it is possible to choose a random spanning tree so that all edges have equal usage probab…
Spanning trees in directed square cycles
本文研究了有向平方环中的生成树结构，推导了其数量公式，并揭示了其与雅各布斯数的关系。
The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current status of the conjecture and provide a further conjecture which implies it.
On stability of weighted spanning tree degree enumerators
本文将生成树度枚举器的实稳定性结果从连通图推广到加权图，并据此定义了加权距离遗传图类。
On the Minimum Spanning Tree Distribution in Grids
本文研究了大网格中最小生成树分布的性质，建立了最可能和最不可能生成树的概率衰减率的界限。
Let T1, T2,..., Tk be spanning trees in a graph G. If for any pair of vertices {u, v} of G, the paths between u and v in every Ti( 0 < i < k+1) do not contain common edges and common vertices, except the vertices u and v, then T1, T2,..., Tk are called completely independent spanning trees in G. The n-dimensional augmented cube, denoted as AQn, a variation of the hypercube possesses several embeddable properties that the hypercube and its variations do not possess. For AQn (n > 5), construction of 4 completely independent spanning trees of which two trees with diameters 2n - 5 and two trees wi…
We give a Cayley type formula to count the number of spanning trees in the complete r-uniform hypergraph for all r >= 3. Similar to the bijection between spanning trees in complete graphs and Parking functions, we derive a bijection from spanning trees of the complete (r+1)-uniform hypergraph which arise from a fixed r-perfect matching and r-Parking functions. We observe a simple consequence of this bijection in terms of the number of regions of the extended Shi arrangement.
We prove a Matrix-Tree Theorem enumerating the spanning trees of a cell complex in terms of the eigenvalues of its cellular Laplacian operators, generalizing a previous result for simplicial complexes. As an application, we obtain explicit formulas for spanning tree enumerators and Laplacian eigenvalues of cubes; the latter are integers. We prove a weighted version of the eigenvalue formula, providing evidence for a conjecture on weighted enumeration of cubical spanning trees. We introduce a cubical analogue of shiftedness, and obtain a recursive formula for the Laplacian eigenvalues of shifte…
We consider all spanning trees of a complete simple graph $Γ$ on $n$ vertices that contain a given $m-$forest $F$. We show that the number of such spanning trees, $τ(F)$, doesn't depend on the structure of $F$ and is completely determined by the number of vertices $q_i \, (i=1, ..., m)$ in each connected component of $F$. Specifically, $τ(F) = q_1 q_2 \cdots q_m n^{m-2}$.
In this paper, we derive closed-form formulas for Kirchhoff index and Wiener index of linear pentagonal cylinder graph and linear pentagonal Mobius chain graph. We also obtain explicit formulas for finding total number of spanning trees for both the graphs.
Balanced Spanning Trees for Triangular Strip Lattices
本文针对三角条带格，通过递推关系和渐近分析，研究了平衡生成树的计数及其比例，并给出了均匀随机选取时平衡生成树的极限概率。
Oriented Minimum spanning tree looks like the Uniform spanning tree on the complete graph (at least locally)
研究完全图中有向最小生成树形图的局部极限，证明其与均匀随机树局部极限相同，与最小生成树不同，揭示引入方向改变最小生成树局部几何结构。
A closure result on spanning $k$-trees of graphs with given minimum degree
研究了具有给定最小度数的图生成k-树的闭包性质，通过顶点度数条件确定图是否可通过添加边转化为生成k-树。
In this paper we construct spanning trees in hyperbolic graphs that represent their hyperbolic compactification in a good way: so that the tree has a bounded number of distinct rays to each boundary point. The bound depends only on the (Assouad) dimension of the boundary. As a corollary we sharpen a result of Gromov which says that from every hyperbolic graph with bounded degrees one can construct a tree outside the graph with a continuous surjection from the ends of the tree onto the hyperbolic boundary such that the surjection is finite-to-one. We will construct a tree with these properties …
Minimal spanning trees on infinite vertex sets are investigated. A criterion for minimality of a spanning tree having a finite length is obtained, which generalizes the corresponding classical result for finite sets. It is given an analytic description of the set of all infinite metric spaces which a minimal spanning tree exists for. A sufficient condition for a minimal spanning tree existence is obtained in terms of distances achievability between partitions elements of the metric space under consideration. Besides, a concept of locally minimal spanning tree is introduced, several properties …
We prove that if a tree $T$ has $n$ vertices and maximum degree at most $Δ$, then a copy of $T$ can almost surely be found in the random graph $\mathcal{G}(n,Δ\log^5 n/n)$.
It is well-known that the number of spanning trees, denoted by $τ(G)$, in a connected multi-graph $G$ can be calculated by the Matrix-Tree theorem and Tutte's deletion-contraction theorem. In this short note, we find an alternate method to compute $τ(G)$ by degrees of vertices.
In this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let $(X,Y)$ be the bipartition of the complete bipartite graph $K_{m,n}$ with $|X|=m$ and $|Y|=n$. We prove that for any given spanning forest $F$ of $K_{m,n}$ with components $T_1,T_2,\ldots,T_k$, the number of spanning trees in $K_{m,n}$ which contain all edges in $F$ is equal to $$ \frac 1{mn}\left(\prod_{i=1}^k (m_in+n_im)\right) \left (1-\sum_{i=1}^{k}\frac{m_in_i}{m_in+n_im}\right ), $$ where $m_i=|V(T_i)\cap X|$ and $n_i=|V(T_i…
An exact-arithmetic algorithm for spanning tree modulus
本文提出一种基于 Cunningham 图脆弱性算法的精确算术算法，用于计算生成树模数及基于强度的层次分解，揭示了模数与临界边集的联系，并通过整数算术实现和扩展性测试验证了其有效性。
The number of spanning trees in a class of directed circulant graphs with generators depending linearly on the number of vertices $βn$, and in the $n$-th and $(n-1)$-th power graphs of the $βn$-cycle are evaluated as a product of $\lceilβ/2\rceil-1$ terms.
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