Average Distance, Independence Number, and Spanning Trees

2013 ◽  
Vol 76 (3) ◽  
pp. 194-199 ◽  
Author(s):  
Simon Mukwembi
Keyword(s):  
Spanning Trees ◽  
Average Distance ◽  
Independence Number

scholarly journals Spanning Trees with Many Leaves and Average Distance

10.37236/757 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ermelinda DeLaViña ◽  
Bill Waller
Keyword(s):  
Lower Bounds ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Average Distance ◽  
Independence Number ◽  
Local Independence ◽  
Open Problems ◽  
Maximum Order ◽  
Bipartite Subgraph ◽  
Number Of Leaves

In this paper we prove several new lower bounds on the maximum number of leaves of a spanning tree of a graph related to its order, independence number, local independence number, and the maximum order of a bipartite subgraph. These new lower bounds were conjectured by the program Graffiti.pc, a variant of the program Graffiti. We use two of these results to give two partial resolutions of conjecture 747 of Graffiti (circa 1992), which states that the average distance of a graph is not more than half the maximum order of an induced bipartite subgraph. If correct, this conjecture would generalize conjecture number 2 of Graffiti, which states that the average distance is not more than the independence number. Conjecture number 2 was first proved by F. Chung. In particular, we show that the average distance is less than half the maximum order of a bipartite subgraph, plus one-half; we also show that if the local independence number is at least five, then the average distance is less than half the maximum order of a bipartite subgraph. In conclusion, we give some open problems related to average distance or the maximum number of leaves of a spanning tree.

DISTANCE PRESERVING SUBTREES IN MINIMUM AVERAGE DISTANCE SPANNING TREES

2013 ◽  
Vol 05 (03) ◽  
pp. 1350010
Author(s):  
LAURENT LYAUDET ◽  
PAULIN MELATAGIA YONTA ◽  
MAURICE TCHUENTE ◽  
RENÉ NDOUNDAM
Keyword(s):  
Approximate Solution ◽  
Spanning Tree ◽  
Path Length ◽  
Spanning Trees ◽  
Undirected Graph ◽  
Average Distance ◽  
Edge Length ◽  
The Other ◽  
First Results ◽  
Positive Length

Given an undirected graph G = (V, E) with n vertices and a positive length w(e) on each edge e ∈ E, we consider Minimum Average Distance (MAD) spanning trees i.e., trees that minimize the path length summed over all pairs of vertices. One of the first results on this problem is due to Wong who showed in 1980 that a Distance Preserving (DP) spanning tree rooted at the median of G is a 2-approximate solution. On the other hand, Dankelmann has exhibited in 2000 a class of graphs where no MAD spanning tree is distance preserving from a vertex. We establish here a new relation between MAD and DP trees in the particular case where the lengths are integers. We show that in a MAD spanning tree of G, each subtree H′ = (V′, E′) consisting of a vertex [Formula: see text] and the union of branches of [Formula: see text] that are each of size less than or equal to [Formula: see text], where w+ is the maximum edge-length in G, is a distance preserving spanning tree of the subgraph of G induced by V′.

ON THE TOTAL DISTANCE AND DIAMETER OF GRAPHS

2018 ◽  
Vol 98 (1) ◽  
pp. 14-17
Author(s):  
HONGBO HUA
Keyword(s):  
Connected Graph ◽  
Spanning Trees ◽  
Average Distance ◽  
Wiener Index ◽  
Connected Graphs ◽  
Total Distance

The total distance (or Wiener index) of a connected graph$G$is the sum of all distances between unordered pairs of vertices of$G$. DeLaViña and Waller [‘Spanning trees with many leaves and average distance’,Electron. J. Combin.15(1) (2008), R33, 14 pp.] conjectured in 2008 that if$G$has diameter$D>2$and order$2D+1$, then the total distance of$G$is at most the total distance of the cycle of the same order. In this note, we prove that this conjecture is true for 2-connected graphs.

On the Location of Quiescent Prominences with Respect to Coronal Holes

1979 ◽  
Vol 44 ◽  
pp. 209-213
Author(s):  
B. Rompolt
Keyword(s):  
Coronal Hole ◽  
Average Distance ◽  
Coronal Holes

The aim of this contribution is to turn attention to a peculiarity of location of the filaments (quiescent prominences) with respect to the boundaries of the coronal holes. It is generally known that quiescent prominences are located at some distance from the boundary of coronal holes. My intention was to check whether the average distance between the nearest border of a coronal hole and the prominence is comparable to the average horizontal extension of a helmet structure overlying the prominence. As well as, whether this average distance depends upon the orientation of the long axis of the prominence with respect to the nearest boundary of the coronal hole.

Mobile network synthesis strategy

2019 ◽  
pp. 98-101
Author(s):  
V. Lyandres
Keyword(s):  
Average Distance ◽  
Mobile Network ◽  
Base Station ◽  
Service Area ◽  
Base Stations ◽  
Frequency Assignment ◽  
Network Synthesis ◽  
Universal Method ◽  
Synthesis Strategy ◽  
Adaptive Vector Quantization

Introduction:Effective synthesis of а mobile communication network includes joint optimisation of two processes: placement of base stations and frequency assignment. In real environments, the well-known cellular concept fails due to some reasons, such as not homogeneous traffic and non-isotropic wave propagation in the service area.Purpose:Looking for the universal method of finding a network structure close to the optimal.Results:The proposed approach is based on the idea of adaptive vector quantization of the network service area. As a result, it is reduced to a 2D discrete map split into zones with approximately equal number of service requests. In each zone, the algorithm finds such coordinates of its base station that provide the shortest average distance to all subscribers. This method takes into account the shortage of the a priory information about the current traffic, ensures maximum coverage of the service area, and what is not less important, significantly simplifies the process of frequency assignment.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

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