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Chung and Graham considered the problem of minimizing the number of edges in an n-vertex graph containing all n-vertex trees as a subgraph. They showed that such a graph has at least 1 2 n log n edges. In this note, we improve this lower estimate ...

In the spanning tree congestion problem, given a connected graph G, the objective is to compute a spanning tree T in G that minimizes its maximum edge congestion, where the congestion of an edge e of T is the number of edges in G for which the ...$\mathbb{}\mathbb{}$$\mathit{}\mathsf{}$$\mathit{}\mathsf{}$$\mathbb{}\mathbb{}$$\mathrm{}$$\mathbf{}\mathsf{}$$\left\{\mathrm{},,\mathrm{},,\mathrm{},,\mathrm{}\right\}$$\mathit{}\mathsf{}$$\mathbb{}\mathbb{}$$\mathrm{}$$\mathrm{}$$\mathit{}\mathsf{}$$\mathbb{}\mathbb{}$$\mathrm{}$

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-...

We study the problem of maximizing the number of full degree vertices in a spanning tree T of a graph G; that is, the number of vertices whose degree in T equals its degree in G. In cubic graphs, this problem is equivalent to maximizing the ...

We prove that a polynomial fraction of the set of k-component forests in the m × n grid graph have equal numbers of vertices in each component, for any constant k. This resolves a conjecture of Charikar, Liu, Liu, and Vuong, and establishes the first ...

A graph whose edges only appear at certain points in time is called a temporal graph (among other names). These graphs are temporally connected if all pairs of vertices are connected by a path that traverses edges in chronological order (a ...

In this paper, we provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs in two different ways: (1) We represent spanning trees and spanning forests in hypergraphs through Berezin-Grassmann ...

• We provide generating functions and counting formulas for spanning trees and spanning forests in hypergraphs through Berezin-Grassmann integrals on Zeon algebra and hyper- Hafnians when orders and signs are not considered.
• We establish ...

Two spanning trees $T_1$ and $T_2$ are called completely independent if there is a path $P_1\in T_1$ from the root node $r_1\in T_1$ to a certain node u that is disjoint with other path $P_2\in T_2$ from the root node $r_2\in T_2$ to a certain node u, where $r_1\ne r_2$. Completely ...

Let N ≥ 2 be an integer, a (1, N)-periodic graph G is a periodic graph whose vertices can be partitioned into two sets V 1 = { v ∣ σ ( v ) = v } and V 2 = { v ∣ σ i ( v ) ≠ v for any 1 < i < N }, where σ is an automorphism with order N of G. The ...

We study the minimum spanning tree cycle intersection (MSTCI) problem on outerplanar graphs in this paper. Consider a connected simple graph G = ( V , E ) and any spanning tree T = ( V , E T ) of G, it is well-known that each non-tree edge e ∈ E ∖...

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.

We show that the spanning tree degree enumerator polynomial of a connected graph G is a real stable polynomial if and only if G is distance-hereditary.

• We enumerate spanning trees with a Kekule structure in the linear hexagonal chains on the plane.
• We enumerate spanning trees with a Kekule structure in the linear hexagonal chains on the cylinder.
• We enumerate spanning trees with a ...

In chemical graph theory, many topological indices of the hexagonal chains, for instance, the energy, Wiener and Kirchhoff indices, and numbers of Kekulé structures and spanning trees, and so on, have been studied extensively. We enumerate ...

• We propose TSRP for UDA. TSRP iteratively exploits the intra-class similarity of the samples in the target domain for improving the generated coarse pseudo ...

Unsupervised domain adaptation (UDA) transfers knowledge from a label-rich source domain to a different but related fully-unlabeled target domain. To address the problem of domain shift, more and more UDA methods adopt pseudo labels of ...

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such ...

As usual λ ( G ) denotes the edge-connectivity of the graph G. It was shown in [2] that every graph G contains a spanning ( λ ( G ) + 1 )-partite subgraph H such that λ ( H ) = λ ( G ) and one can find such a spanning subgraph in ...

In this paper, we derive closed-form formulas for the Kirchhoff index and Wiener index of cylinder pentagonal chain and Möbius pentagonal chain. We also obtain explicit formulas for finding the total number of spanning trees for both ...

A subgraph $H=(V,E^{\prime })$ of a graph $G=(V,E)$ is non-separating if $G\backslash E^{\prime }$, that is, the graph obtained from G by deleting the edges in $E^{\prime }$, is connected. Analogously we say that a subdigraph $X=(V,A^{\prime })$ of a digraph $D=(V,A)$ is non-separating if $D\backslash A^{\prime }$ is ...$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$

Let $T_1,T_2,\dots ,T_k$ be $k(\ge 2)$ spanning trees of a graph G. The trees $T_1,T_2,\dots ,T_k$ are completely independent if the paths connecting any two vertices of G in these k trees are pairwise internally disjoint. Sufficient conditions for a graph to possess k ...$$$$$\mathrm{}$$$$\frac{\mathrm{}}{}\mathrm{}\frac{\mathrm{}}{}\mathrm{}$$$$\mathrm{}\mathrm{}$$\mathrm{}$$\frac{\mathrm{}{\mathrm{}}^{\mathrm{}}\mathrm{}}{\mathrm{}{\mathrm{}}^{\mathrm{}}}\mathrm{}$$\frac{\mathrm{}}{\mathrm{}}\mathrm{}$$$

In a reconfiguration problem, given a problem and two feasible solutions of the problem, the task is to find a sequence of transformations to reach from one solution to the another such that every intermediate state is also a feasible solution to ...$$