scholarly journals The number of triangles is more when they have no common vertex

2021 ◽  
Vol 344 (5) ◽  
pp. 112330
Author(s):  
Chuanqi Xiao ◽  
Gyula O.H. Katona
Keyword(s):  
Common Vertex

scholarly journals Finding Optimal 1-Endpoint-Crossing Trees

2013 ◽  
Vol 1 ◽  
pp. 13-24 ◽  
Author(s):  
Emily Pitler ◽  
Sampath Kannan ◽  
Mitchell Marcus
Keyword(s):  
Natural Language ◽  
Common Vertex ◽  
Dependency Parsing ◽  
Dependency Tree ◽  
Parsing Algorithm

Dependency parsing algorithms capable of producing the types of crossing dependencies seen in natural language sentences have traditionally been orders of magnitude slower than algorithms for projective trees. For 95.8–99.8% of dependency parses in various natural language treebanks, whenever an edge is crossed, the edges that cross it all have a common vertex. The optimal dependency tree that satisfies this 1-Endpoint-Crossing property can be found with an O( n4) parsing algorithm that recursively combines forests over intervals with one exterior point. 1-Endpoint-Crossing trees also have natural connections to linguistics and another class of graphs that has been studied in NLP.

scholarly journals Experimental and Numerical Analysis of Additively Manufactured Inconel 718 Coupons With Lattice Structure

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Sarwesh Parbat ◽  
Zheng Min ◽  
Li Yang ◽  
Minking Chyu
Keyword(s):  
Heat Transfer ◽  
Steady State ◽  
Pressure Drop ◽  
Inconel 718 ◽  
Lattice Structure ◽  
Unit Cell ◽  
Common Vertex ◽  
Constant Wall Temperature ◽  
Near Surface ◽  
Smooth Channel

Abstract In the present paper, two lattice geometries suitable for near surface and double wall cooling were developed and tested. The first type of unit cell consisted of six ligaments of 0.5 mm diameter joined at a common vertex near the middle. The second type of unit cell was derived from the first type by adding four mutually perpendicular ligaments in the middle plane. Two lattice configurations, referred to as L1 and L2, respectively, were obtained by repeating the corresponding unit cell in streamwise and spanwise directions in an inline fashion. Test coupons consisting of these lattice geometries embedded inside rectangular cooling channel with dimensions of 2.54 mm height, 38.07 mm width, and 38.1 mm in length were fabricated using Inconel 718 powder and selective laser sintering (SLS) process. The heat transfer and pressure drop performance was then evaluated using steady-state tests with constant wall temperature boundary condition and for channel Reynolds number ranging from 2800 to 15,000. The lattices depicted a higher heat transfer compared with a smooth channel and both the heat transfer and pressure drop increased with a decrease in the porosity from L1 to L2. Steady-state conjugate numerical results revealed formation of prominent vortical structures in the inter-unit cell spaces, which diverted the flow toward the top end wall and created an asymmetric heat transfer between the two end walls. In conclusion, these lattice structures provided an augmented heat transfer while favorably redistributing the coolant within channel.

Generation of Surface Models of 3D Objects From Their Wire-Frame Models

1992 ◽  
Author(s):  
S. Gupta ◽  
A. Shirkhodaie ◽  
A. H. Soni
Keyword(s):  
Common Vertex ◽  
Surface Model ◽  
Edge Information ◽  
Frame Model ◽  
3D Objects ◽  
Wire Frame ◽  
Surface Models ◽  
Plane Passing ◽  
Generate Surface ◽  
The Wire

Abstract This paper presents an algorithm to generate surface models of 3D objects from their wire-frame models. The algorithm firstly, obtains information about edges of the object from the wire-frame model of the object and uses this edge information to generate the pairs. A pair of an object is a combination of two non-collinear edges which have a common vertex. The algorithm then determines the unique plane passing through each pair and groups the coplanar pairs together. Then it sorts each of the groups of coplanar pairs to form one or more loops of edges. Finally for each group of coplanar pairs, all the loops are combined, using a few rules, to form faces of the object. Hence a surface model of the object is generated.

scholarly journals Efficient Algorithm for Generating Maximal L-Reflexive Trees

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 809
Author(s):  
Milica Anđelić ◽  
Dejan Živković
Keyword(s):  
Efficient Algorithm ◽  
Line Graph ◽  
Common Vertex ◽  
Line Graphs ◽  
Algebraic Numbers ◽  
Pisot Numbers ◽  
Full Characterization ◽  
Vertex Set ◽  
Second Largest Eigenvalue

The line graph of a graph G is another graph of which the vertex set corresponds to the edge set of G, and two vertices of the line graph of G are adjacent if the corresponding edges in G share a common vertex. A graph is reflexive if the second-largest eigenvalue of its adjacency matrix is no greater than 2. Reflexive graphs give combinatorial ground to generate two classes of algebraic numbers, Salem and Pisot numbers. The difficult question of identifying those graphs whose line graphs are reflexive (called L-reflexive graphs) is naturally attacked by first answering this question for trees. Even then, however, an elegant full characterization of reflexive line graphs of trees has proved to be quite formidable. In this paper, we present an efficient algorithm for the exhaustive generation of maximal L-reflexive trees.

A Note on Primitive Graphs

1972 ◽  
Vol 15 (3) ◽  
pp. 437-440 ◽  
Author(s):  
I. Z. Bouwer ◽  
G. F. LeBlanc
Keyword(s):  
Connected Graph ◽  
Common Vertex ◽  
Connected Subgraph ◽  
Property A ◽  
Vertex Set

Let G denote a connected graph with vertex set V(G) and edge set E(G). A subset C of E(G) is called a cutset of G if the graph with vertex set V(G) and edge set E(G)—C is not connected, and C is minimal with respect to this property. A cutset C of G is simple if no two edges of C have a common vertex. The graph G is called primitive if G has no simple cutset but every proper connected subgraph of G with at least one edge has a simple cutset. For any edge e of G, let G—e denote the graph with vertex set V(G) and with edge set E(G)—e.

scholarly journals The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 125
Author(s):  
Ismael González Yero
Keyword(s):  
Vertex Cover ◽  
Common Vertex ◽  
Metric Dimension ◽  
Minimum Cardinality ◽  
New Approach ◽  
Strong Metric Dimension ◽  
Vertex Set ◽  
Graph Families ◽  
Vertex Sets ◽  
Np Hardness

We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V ( G ) , and the following terminology. Two vertices u , v ∈ V ( G ) are strongly resolved by a vertex w ∈ V ( G ) , if there is a shortest w − v path containing u or a shortest w − u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S ⊂ V is an SSMG for F , if such set S is a strong metric generator for every graph G ∈ F . The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F , and is denoted by Sd s ( F ) . The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sd s ( F ) is described. That is, it is proved that computing Sd s ( F ) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F . Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature.

A New Agglomeration Method for Isotropic Unstructured Grids

2012 ◽  
Vol 241-244 ◽  
pp. 2957-2961
Author(s):  
Zong Zhe Li ◽  
Zheng Hua Wang ◽  
Wei Cao ◽  
Lu Yao
Keyword(s):  
Aspect Ratio ◽  
Euler Equations ◽  
Multigrid Method ◽  
Unstructured Grid ◽  
Convergence Rates ◽  
Unstructured Grids ◽  
Common Vertex ◽  
Finite Volume Scheme ◽  
High Quality ◽  
Coarse Grids

A robust aspect ratio based agglomeration algorithm to generate high quality coarse grids for unstructured grid is proposed in this paper. The algorithm focuses on multigrid techniques for the numerical solution of Euler equations, which conform to cell-centered finite volume scheme, combines isotropic vertex-based agglomeration to yield large increases in convergence rates. Aspect ratio is used as fusing weight to capture the degree of cell convexity and give an indication of cell quality, agglomerating isotropic cells sharing a common vertex. Consequently, we conduct agglomeration multigrid method to solve Euler equations on 2D isotropic unstructured grid, and compare the results with MGridGen

On the Arithmetical Rank of the Edge Ideals of Some Graphs

2012 ◽  
Vol 19 (spec01) ◽  
pp. 797-806 ◽  
Author(s):  
Fatemeh Mohammadi ◽  
Dariush Kiani
Keyword(s):  
Projective Dimension ◽  
Monomial Ideals ◽  
Common Vertex ◽  
Edge Ideals ◽  
Arithmetical Rank

In this paper, we compute the projective dimension of the edge ideals of graphs consisting of some cycles and lines which are joint in a common vertex. Moreover, we show that for such graphs, the arithmetical rank equals the projective dimension. As an application, we can compute the arithmetical rank for some homogenous monomial ideals.

Trees with given maximum degree minimizing the spectral radius

2016 ◽  
Vol 31 ◽  
pp. 335-361
Author(s):  
Xue Du ◽  
Lingsheng Shi
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Maximum Degree ◽  
Equal Length ◽  
Disjoint Paths ◽  
Common Vertex ◽  
Largest Eigenvalue ◽  
Large N ◽  
The Largest Eigenvalue

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let $T^*(n,\Delta ,l)$ be the tree which minimizes the spectral radius of all trees of order $n$ with exactly $l$ vertices of maximum degree $\Delta $. In this paper, $T^*(n,\Delta ,l)$ is determined for $\Delta =3$, and for $l\le 3$ and $n$ large enough. It is proven that for sufficiently large $n$, $T^*(n,3,l)$ is a caterpillar with (almost) uniformly distributed legs, $T^*(n,\Delta ,2)$ is a dumbbell, and $T^*(n,\Delta ,3)$ is a tree consisting of three distinct stars of order $\Delta $ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

scholarly journals Oriented Tait graphs

1973 ◽  
Vol 16 (3) ◽  
pp. 328-331 ◽  
Author(s):  
G. Szekeres
Keyword(s):  
Planar Graph ◽  
Common Vertex ◽  
Edge Disjoint ◽  
Trivalent Graph ◽  
Edge Colouring

The four colour conjecture is well known to be equivalent to the proposition that every trivalent planar graph without an isthmus (i.e. an edge whose removal disconnects the graph) has an edge colouring in three colours ([1], p. 121). By an edge colouring we mean an assignment of colours to the edges of the graph so that no two edges of the same colour meet at a common vertex, and the graph is n-valent if n edges meet at each vertex. An edge colouring by three colours is called a Tait colouring; a trivalent graph which has a Tait colouring can be split in three edge-disjoint 1-factors, i.e. spanning monovalent subgraphs.

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