On the metric dimension of corona product graphs

Given a set of vertices $S=\{v_1,v_2,...,v_k\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\in \{1,...,k\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\ne r(v|S)$. The metric dimension of $G$, $dim(G)$, is the minimum cardinality of any resolving set for $G$. Let $G$ and $H$ be two graphs of order $n_1$ and $n_2$, respectively. The corona product $G\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n_1$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. For any integer $k\ge 2$, we define the graph $G\odot^k H$ recursively from $G\odot H$ as $G\odot^k H=(G\odot^{k-1} H)\odot H$. We give several results on the metric dimension of $G\odot^k H$. For instance, we show that given two connected graphs $G$ and $H$ of order $n_1\ge 2$ and $n_2\ge 2$, respectively, if the diameter of $H$ is at most two, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(H)$. Moreover, if $n_2\ge 7$ and the diameter of $H$ is greater than five or $H$ is a cycle graph, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(K_1\odot H).

Spiritual formation of millennials : an exploration of best practices for Crosspoint Church

https://place.asburyseminary.edu/ecommonsatsdissertations/2457/thumbnail.jp

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most $k$. Ban and Linial conjectured that every bridgeless cubic graph admits a $2$-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph $G$ with $|E(G)| \equiv 0 \pmod 2$ has a $2$-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying $2$-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.Comment: 33 pages; submitted for publicatio

Spectral Measures of Bipartivity in Complex Networks

We introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network. Spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity. It is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks, trophic interactions in food webs, construction principles in metabolic networks, or communities in social networks.Comment: 16 pages, 1 figure, 1 tabl

Operational Improvements From the Automatic Dependant Surveillance Broadcast In-Trail Procedure in the Pacific Organized Track System

The Federal Aviation Administration's Surveillance and Broadcast Services Program has supported implementation of the Automatic Dependant Surveillance Broadcast (ADS-B) In-Trail Procedure (ITP) on commercial revenue flights. ADS-B ITP is intended to be used in non-radar airspace that is employing procedural separation. Through the use of onboard tools, pilots are able to make a new type of altitude change request to an Air Traffic Service Provider (ATSP). The FAA, in partnership with United Airlines, is conducting flight trials of the ITP in revenue service in the Pacific. To support the expansion of flight trials to the rest of the US managed Pacific Airspace Region, a computerized batch study was conducted to investigate the operational impacts and potential benefits that can be gained through the use of the ITP in the Pacific Organized Track System (PACOTS). This study, which simulated the Oakland managed portion of the PACOTS, suggests that potential benefits in the PACOTS are significant with a considerable increase in time spent at optimum altitude and associated fuel savings

Resolving domination in graphs

summary:For an ordered set $W =\lbrace w_1, w_2, \cdots , w_k\rbrace$ of vertices and a vertex $v$ in a connected graph $G$, the (metric) representation of $v$ with respect to $W$ is the $k$-vector $r(v|W) = (d(v, w_1),d(v, w_2) ,\cdots , d(v, w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for $G$ is its dimension $\dim G$. A set $S$ of vertices in $G$ is a dominating set for $G$ if every vertex of $G$ that is not in $S$ is adjacent to some vertex of $S$. The minimum cardinality of a dominating set is the domination number $\gamma (G)$. A set of vertices of a graph $G$ that is both resolving and dominating is a resolving dominating set. The minimum cardinality of a resolving dominating set is called the resolving domination number $\gamma _r(G)$. In this paper, we investigate the relationship among these three parameters

Divergence in Dialogue

Copyright: 2014 Healey et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.This work was supported by the Economic and Social Research Council (ESRC; http://www.esrc.ac.uk/) through the DynDial project (Dynamics of Conversational Dialogue, RES-062-23-0962) and the Engineering and Physical Sciences Research Council (EPSRC; http://www.epsrc.ac.uk/) through the RISER project (Robust Incremental Semantic Resources for Dialogue, EP/J010383/1). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields

In this paper we discuss off-shell representations of N-extended supersymmetry in one dimension, ie, N-extended supersymmetric quantum mechanics, and following earlier work on the subject codify them in terms of certain graphs, called Adinkras. This framework provides a method of generating all Adinkras with the same topology, and so also all the corresponding irreducible supersymmetric multiplets. We develop some graph theoretic techniques to understand these diagrams in terms of a relatively small amount of information, namely, at what heights various vertices of the graph should be "hung". We then show how Adinkras that are the graphs of N-dimensional cubes can be obtained as the Adinkra for superfields satisfying constraints that involve superderivatives. This dramatically widens the range of supermultiplets that can be described using the superspace formalism and organizes them. Other topologies for Adinkras are possible, and we show that it is reasonable that these are also the result of constraining superfields using superderivatives. The family of Adinkras with an N-cubical topology, and so also the sequence of corresponding irreducible supersymmetric multiplets, are arranged in a cyclical sequence called the main sequence. We produce the N=1 and N=2 main sequences in detail, and indicate some aspects of the situation for higher N.Comment: LaTeX, 58 pages, 52 illustrations in color; minor typos correcte

On the Metric Dimension of Cartesian Products of Graphs

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded

SURF IA Conflict Detection and Resolution Algorithm Evaluation

The Enhanced Traffic Situational Awareness on the Airport Surface with Indications and Alerts (SURF IA) algorithm was evaluated in a fast-time batch simulation study at the National Aeronautics and Space Administration (NASA) Langley Research Center. SURF IA is designed to increase flight crew situation awareness of the runway environment and facilitate an appropriate and timely response to potential conflict situations. The purpose of the study was to evaluate the performance of the SURF IA algorithm under various runway scenarios, multiple levels of conflict detection and resolution (CD&R) system equipage, and various levels of horizontal position accuracy. This paper gives an overview of the SURF IA concept, simulation study, and results. Runway incursions are a serious aviation safety hazard. As such, the FAA is committed to reducing the severity, number, and rate of runway incursions by implementing a combination of guidance, education, outreach, training, technology, infrastructure, and risk identification and mitigation initiatives [1]. Progress has been made in reducing the number of serious incursions - from a high of 67 in Fiscal Year (FY) 2000 to 6 in FY2010. However, the rate of all incursions has risen steadily over recent years - from a rate of 12.3 incursions per million operations in FY2005 to a rate of 18.9 incursions per million operations in FY2010 [1, 2]. The National Transportation Safety Board (NTSB) also considers runway incursions to be a serious aviation safety hazard, listing runway incursion prevention as one of their most wanted transportation safety improvements [3]. The NTSB recommends that immediate warning of probable collisions/incursions be given directly to flight crews in the cockpit [4]