SOME CARTESIAN PRODUCTS OF A PATH AND PRISM RELATED GRAPHS THAT ARE EDGE ODD GRACEFUL

Let $G$ be a connected undirected simple graph of size $q$ and let $k$ be the maximum number of its order and its size. Let $f$ be a bijective edge labeling which codomain is the set of odd integers from 1 up to $2q-1$. Then $f$ is called an edge odd graceful on $G$ if the weights of all vertices are distinct, where the weight of a vertex $v$ is defined as the sum $mod(2k)$ of all labels of edges incident to $v$. Any graph that admits an edge odd graceful labeling is called an edge odd graceful graph. In this paper, some new graph classes that are edge odd graceful are presented, namely some cartesian products of path of length two and some circular related graphs

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

International Journal of Mathematical Combinatorics, Vol.6

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Games on graphs, visibility representations, and graph colorings

In this thesis we study combinatorial games on graphs and some graph parameters whose consideration was inspired by an interest in the symmetry of hypercubes. A capacity function f on a graph G assigns a nonnegative integer to each vertex of V(G). An f-matching in G is a set M ⊆ E(G) such that the number of edges of M incident to v is at most f(v) for all v ⊆ V(G). In the f-matching game on a graph G, denoted (G,f), players Max and Min alternately choose edges of G to build an f-matching; the game ends when the chosen edges form a maximal f-matching. Max wants the final f-matching to be large; Min wants it to be small. The f-matching number is the size of the final f-matching under optimal play. We extend to the f-matching game a lower bound due to Cranston et al. on the game matching number. We also consider a directed version of the f-matching game on a graph G. Peg Solitaire is a game on connected graphs introduced by Beeler and Hoilman. In the game, pegs are placed on all but one vertex. If x, y, and z form a 3-vertex path and x and y each have a peg but z does not, then we can remove the pegs at x and y and place a peg at z; this is called a jump. The goal of the Peg Solitaire game on graphs is to find jumps that reduce the number of pegs on the graph to 1. Beeler and Rodriguez proposed a variant where we want to maximize the number of pegs remaining when no more jumps can be made. Maximizing over all initial locations of a single hole, the maximum number of pegs left on a graph G when no jumps remain is the Fool's Solitaire number F(G). We determine the Fool's Solitaire number for the join of any graphs G and H. For the cartesian product, we determine F(G ◻ K_k) when k ≥ 3 and G is connected. Finally, we give conditions on graphs G and H that imply F(G ◻ H) ≥ F(G) F(H). A t-bar visibility representation of a graph G assigns each vertex a set that is the union of at most t horizontal segments ("bars") in the plane so that vertices are adjacent if and only if there is an unobstructed vertical line of sight (having positive width) joining the sets assigned to them. The visibility number of a graph G, written b(G), is the least t such that G has a t-bar visibility representation. Let Q_n denote the n-dimensional hypercube. A simple application of Euler's Formula yields b(Q_n) ≥ ⌈(n+1)/4⌉. To prove that equality holds, we decompose Q_{4k-1} explicitly into k spanning subgraphs whose components have the form C_4 ◻ P_{2^l}. The visibility number b(D) of a digraph D is the least t such that D can be represented by assigning each vertex at most t horizontal bars in the plane so that uv ∈ E(D) if and only if there is an unobstructed vertical line of sight (with positive width) joining some bar for u to some higher bar for v. It is known that b(D) ≤ 2 for every outerplanar digraph. We give a characterization of outerplanar digraphs with b(D)=1. A proper vertex coloring of a graph G is r-dynamic if for each v ∈ V (G), at least min{r, d(v)} colors appear in N_G(v). We investigate r-dynamic versions of coloring and list coloring. We give upper bounds on the minimum number of colors needed for any r in terms of the genus of the graph. Two vertices of Q_n are antipodal if they differ in every coordinate. Two edges uv and xy are antipodal if u is antipodal to x and v is antipodal to y. An antipodal edge-coloring of Q_n is a 2-coloring of the edges in which antipodal edges have different colors. DeVos and Norine conjectured that for n ≥ 2, in every antipodal edge-coloring of Q_n there is a pair of antipodal vertices connected by a monochromatic path. Previously this was shown for n ≤ 5. Here we extend this result to n = 6. Hovey introduced A-cordial labelings as a simultaneous generalization of cordial and harmonious labelings. If S is an abelian group, then a labeling f: V(G) → A of the vertices of a graph G induces an edge-labeling on G; the edge uv receives the label f(u) + f(v). A graph G isA-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most 1, and (2) the induced edge label classes differ in size by at most 1. The smallest non-cyclic group is V_4 (also known as Z_2×Z_2). We investigate V_4-cordiality of many families of graphs, namely complete bipartite graphs, paths, cycles, ladders, prisms, and hypercubes. Finally, we introduce a generalization of A-cordiality involving digraphs and quasigroups, and we show that there are infinitely many Q-cordial digraphs for every quasigroup Q

Visualizing Set Relations and Cardinalities Using Venn and Euler Diagrams

In medicine, genetics, criminology and various other areas, Venn and Euler diagrams are used to visualize data set relations and their cardinalities. The data sets are represented by closed curves and the data set relationships are depicted by the overlaps between these curves. Both the sets and their intersections are easily visible as the closed curves are preattentively processed and form common regions that have a strong perceptual grouping effect. Besides set relations such as intersection, containment and disjointness, the cardinality of the sets and their intersections can also be depicted in the same diagram (referred to as area-proportional) through the size of the curves and their overlaps. Size is a preattentive feature and so similarities, differences and trends are easily identified. Thus, such diagrams facilitate data analysis and reasoning about the sets. However, drawing these diagrams manually is difficult, often impossible, and current automatic drawing methods do not always produce appropriate diagrams. This dissertation presents novel automatic drawing methods for different types of Euler diagrams and a user study of how such diagrams can help probabilistic judgement. The main drawing algorithms are: eulerForce, which uses a force-directed approach to lay out Euler diagrams; eulerAPE, which draws area-proportional Venn diagrams with ellipses. The user study evaluated the effectiveness of area- proportional Euler diagrams, glyph representations, Euler diagrams with glyphs and text+visualization formats for Bayesian reasoning, and a method eulerGlyphs was devised to automatically and accurately draw the assessed visualizations for any Bayesian problem. Additionally, analytic algorithms that instantaneously compute the overlapping areas of three general intersecting ellipses are provided, together with an evaluation of the effectiveness of ellipses in drawing accurate area-proportional Venn diagrams for 3-set data and the characteristics of the data that can be depicted accurately with ellipses

Part decomposition of 3D surfaces

This dissertation describes a general algorithm that automatically decomposes realworld scenes and objects into visual parts. The input to the algorithm is a 3 D triangle mesh that approximates the surfaces of a scene or object. This geometric mesh completely specifies the shape of interest. The output of the algorithm is a set of boundary contours that dissect the mesh into parts where these parts agree with human perception. In this algorithm, shape alone defines the location of a bom1dary contour for a part. The algorithm leverages a human vision theory known as the minima rule that states that human visual perception tends to decompose shapes into parts along lines of negative curvature minima. Specifically, the minima rule governs the location of part boundaries, and as a result the algorithm is known as the Minima Rule Algorithm. Previous computer vision methods have attempted to implement this rule but have used pseudo measures of surface curvature. Thus, these prior methods are not true implementations of the rule. The Minima Rule Algorithm is a three step process that consists of curvature estimation, mesh segmentation, and quality evaluation. These steps have led to three novel algorithms known as Normal Vector Voting, Fast Marching Watersheds, and Part Saliency Metric, respectively. For each algorithm, this dissertation presents both the supporting theory and experimental results. The results demonstrate the effectiveness of the algorithm using both synthetic and real data and include comparisons with previous methods from the research literature. Finally, the dissertation concludes with a summary of the contributions to the state of the art

Calibration of Flush Air Data Sensing Systems Using Surrogate Modeling Techniques

In this work the problem of calibrating Flush Air Data Sensing (FADS) has been addressed. The inverse problem of extracting freestream wind speed and angle of attack from pressure measurements has been solved. The aim of this work was to develop machine learning and statistical tools to optimize design and calibration of FADS systems. Experimental and Computational Fluid Dynamics (EFD and CFD) solve the forward problem of determining the pressure distribution given the wind velocity profile and bluff body geometry. In this work three ways are presented in which machine learning techniques can improve calibration of FADS systems. First, a scattered data approximation scheme, called Sequential Function Approximation (SFA) that successfully solved the current inverse problem was developed. The proposed scheme is a greedy and self-adaptive technique that constructs reliable and robust estimates without any user-interaction. Wind speed and direction prediction algorithms were developed for two FADS problems. One where pressure sensors are installed on a surface vessel and the other where sensors are installed on the Runway Assisted Landing Site (RALS) control tower. Second, a Tikhonov regularization based data-model fusion technique with SFA was developed to fuse low fidelity CFD solutions with noisy and sparse wind tunnel data. The purpose of this data model fusion approach was to obtain high fidelity, smooth and noiseless flow field solutions by using only a few discrete experimental measurements and a low fidelity numerical solution. This physics based regularization technique gave better flow field solutions compared to smoothness based solutions when wind tunnel data is sparse and incomplete. Third, a sequential design strategy was developed with SFA using Active Learning techniques from the machine learning theory and Optimal Design of Experiments from statistics for regression and classification problems. Uncertainty Sampling was used with SFA to demonstrate the effectiveness of active learning versus passive learning on a cavity flow classification problem. A sequential G-optimal design procedure was also developed with SFA for regression problems. The effectiveness of this approach was demonstrated on a simulated problem and the above mentioned FADS problem