Related papers: Counting Isomorphism Classes of Spanning Trees of …
The number of spanning trees in a graph $G$ is the total number of distinct spanning subgraphs of $G$ that are trees. In this paper we characterize the unique graph with a prescribed vertex (resp. edge) connectivity, minimum degree and…
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…
Using the theory of electrical network, we first obtain a simple formula for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we apply the effective resistance (i.e.,…
Let $R$ and $B$ be two disjoint sets of points in the plane such that $|B|\leqslant |R|$, and no three points of $R\cup B$ are collinear. We show that the geometric complete bipartite graph $K(R,B)$ contains a non-crossing spanning tree…
In this paper, we first prove that if the edges of $K_{2m}$ are properly colored by $2m-1$ colors in such a way that any two colors induce a 2-factor of which each component is a 4-cycle, then $K_{2m}$ can be decomposed into $m$ isomorphic…
We prove that every connected graph with $s$ vertices of degree~1 and 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${1\over 3}t +{1\over 4}s+{3\over 2}$ leaves. We present infinite series of graphs showing that…
We prove, that every connected graph with $s$ vertices of degree 3 and $t$ vertices of degree at least~4 has a spanning tree with at least ${2\over 5}t +{1\over 5}s+\alpha$ leaves, where $\alpha \ge {8\over 5}$. Moreover, $\alpha \ge 2$ for…
In this paper we examine the classes of graphs whose $K_n$-complements are trees and quasi-threshold graphs and derive formulas for their number of spanning trees; for a subgraph $H$ of $K_n$, the $K_n$-complement of $H$ is the graph…
We prove that every connected graph with $s$ vertices of degree not 2 has a spanning tree with at least ${1\over 4}(s-2)+2$ leaves. Let $G$ be a be a connected graph of girth $g$ with $v>1$ vertices. Let maximal chain of successively…
A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite…
Recently, the second and third author showed that complete geometric graphs on $2n$ vertices in general cannot be partitioned into $n$ plane spanning trees. Building up on this work, in this paper, we initiate the study of partitioning into…
We consider lower bounds on the number of spanning trees of connected graphs with degree bounded by $d$. The question is of interest because such bounds may improve the analysis of the improvement produced by memorisation in the runtime of…
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…
We compute the total number of spanning trees for the generalized cone of the complete graph $K_n$ and a number of families of some modified bipartite graphs $K_{m,n}$. In particular, we obtain a new method of finding the number of spanning…
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…
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…
In this paper, we address the Ehrenborg's conjecture which proposes that for any bipartite graph the number of spanning trees does not exceed the product of the degrees of the vertices divided by the product of the sizes of the graph…
A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood…
We characterize all partitions of the complete twisted graph $T_{2n}$ into plane spanning trees. In the case of partitions of $T_{2n}$ into isomorphic plane spanning trees, we show that all trees in these partitions must be balanced double…
In this paper, we introduce two families of planar and self-similar graphs which have small-world properties. The constructed models are based on an iterative process where each step of a certain formulation of modules results in a final…