Related papers: Counting Spanning Trees of Threshold Graphs
We give combinatorial proofs of some enumeration formulas involving labelled threshold, quasi-threshold, loop-threshold and quasi-loop-threshold graphs. In each case we count by number of vertices and number of components. For threshold…
We show an $\widetilde{O}(m^{1.5} \epsilon^{-1})$ time algorithm that on a graph with $m$ edges and $n$ vertices outputs its spanning tree count up to a multiplicative $(1+\epsilon)$ factor with high probability, improving on the previous…
A threshold graph is any graph which can be constructed from the empty graph by repeatedly adding a new vertex that is either adjacent to every vertex or to no vertices. The Eulerian number $\genfrac{\langle}{\rangle}{0pt}{}{n}{k}$ counts…
A new very simple proof of the number of labeled rooted forest-graphs with a given number of vertices is given. As a partial case of this formula we have Cayley's formula.
This paper studies increasing trees on $n$ labeled vertices, in which labels increase from the root to the leaves. It is known that the number of binary increasing trees coincides with the number of alternating permutations (Euler numbers).…
In this paper, we provide algorithms to rank and unrank certain degree-restricted classes of Cayley trees (spanning trees of the n-vertex complete graph). Specifically, we consider classes of trees that have a given set of leaves or a fixed…
A number which is either the square of an integer or two times the square of an integer is called squarish. There are two main results in the literature on graphs whose number of perfect matchings is squarish: one due to Jockusch (for…
In a graph, a spanning tree is said to be a tree t-spanner of the graph if the distance between any two vertices in is at most times their distance in . The tree t-spanner has many applications in networks and distributed environments. In…
Let P be a set of n > 2 points in general position in the plane and let G be a geometric graph with vertex set P. If the number of empty triangles uvw in P for which the subgraph of G induced by {u,v,w} is not connected is at most n-3, then…
While the notion of arboricity of a graph is well-known in graph theory, very few results are dedicated to the minimal number of trees covering the edges of a graph, called the tree number of a graph.
We show that every connected $k$-chromatic graph contains at least $k^{k-2}$ spanning trees.
We prove a weighted generalization of the formula for the number of plane vertex-labeled trees.
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…
This paper studies the relation among the number of spanning trees of intermediate graphs in a Galois cover, building on results for $(\mathbb{Z}/2\mathbb{Z})^m$-covers previously established by Hammer, Mattman, Sands, and Valli\`{e}res. We…
Let $R$ and $B$ be two disjoint sets of points in the plane where the points of $R$ are colored red and the points of $B$ are colored blue, and let $n=|R\cup B|$. A bichromatic spanning tree is a spanning tree in the complete bipartite…
The line graph LG of a directed graph G has a vertex for every edge of G and an edge for every path of length 2 in G. In 1967, Knuth used the Matrix-Tree Theorem to prove a formula for the number of spanning trees of LG, and he asked for a…
The degree sequence of a graph is a numerical method to characterize the properties of graphs. Generalized forms of degree sequences exist for complete graphs and complete graphs. Nikolopolus et al. characterized the number of spanning…
A problem of practical and theoretical interest is to determine or estimate the diameter of various families of Cayley networks. The previously known estimate for the diameter of Cayley graphs generated by transposition trees is an upper…
In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the…
Over some types of trees with a given number of vertices, which trees minimize or maximize the total number of subtrees or leaf containing subtrees are studied. Here are some of the main results:\ (1)\, Sharp upper bound on the total number…