AI & ChatGPT searches , social queriess for BOUND GRAPH
Search references for BOUND GRAPH. Phrases containing BOUND GRAPH
See searches and references containing BOUND GRAPH!
Search references for BOUND GRAPH. Phrases containing BOUND GRAPH
See searches and references containing BOUND GRAPH!BOUND GRAPH
Concept in graph theory
graph theory, a bound graph expresses which pairs of elements of some partially ordered set have an upper bound. Rigorously, any graph G is a bound graph
Bound_graph
Methodic assignment of colors to elements of a graph
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a graph. The assignment is subject to certain
Graph_coloring
Sparse graph with strong connectivity
In graph theory, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion. Expander
Expander_graph
Writing paper with a grid
regular grid. It is available either as loose leaf paper or bound in notebooks or graph books. It is commonly found in mathematics and engineering education
Graph_paper
Computer science algorithm
computer science, graph traversal (also known as graph search) refers to the process of visiting (checking and/or updating) each vertex in a graph. Such traversals
Graph_traversal
Linear algebra aspects of graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors
Spectral_graph_theory
Graph with all vertices of degree 3
of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are
Cubic_graph
Square matrix used to represent a graph or network
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether
Adjacency_matrix
Graph that can be embedded in the plane
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect
Planar_graph
Area of discrete mathematics
computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context
Graph_theory
Unsolved problem in computational complexity theory
Bounded-parameter graphs Graphs of bounded treewidth Graphs of bounded genus (Planar graphs are graphs of genus 0.) Graphs of bounded degree Graphs with bounded eigenvalue
Graph_isomorphism_problem
Appendix:Glossary of graph theory in Wiktionary, the free dictionary. This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes
Glossary_of_graph_theory
Adjacent subset of an undirected graph
lower bound on the size of a clique in dense graphs. If a graph has sufficiently many edges, it must contain a large clique. For instance, every graph with
Clique_(graph_theory)
Spectral graph theory concept
spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are
Ramanujan_graph
Measure of capacity of a communications channel defined from a graph
Shannon capacity of a communications channel defined from the graph, and is upper bounded by the Lovász number, which can be computed in polynomial time
Shannon_capacity_of_a_graph
Geometric graph with unit edge lengths
distance graph on n {\displaystyle n} vertices. The best known upper bound is O ( n 4 / 3 ) . {\displaystyle O(n^{4/3}).} The best known lower bound is Ω
Unit_distance_graph
Longest distance between two vertices
interval graphs, and in near-linear time for graphs of bounded treewidth. In median graphs, the diameter can be found in the subquadratic time bound O ~ (
Diameter_(graph_theory)
Form of data structure
A scene graph is a hierarchical data structure commonly used by vector-based graphics editing applications and modern computer games, which cascades the
Scene_graph
Class of artificial neural networks
Graph neural networks (GNNs) are artificial neural networks designed for tasks whose inputs are graphs. Because graphs usually do not have a canonical
Graph_neural_network
Directed graph with no directed cycles
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it
Directed_acyclic_graph
Task of computing complete subgraphs
families of graphs in which the number of cliques is polynomially bounded. These families include chordal graphs, complete graphs, triangle-free graphs, interval
Clique_problem
Statement in mathematical combinatorics
its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. As
Ramsey's_theorem
Regular graph with girth more than twice its diameter
_{i=0}^{k-1}(d-1)^{i},} an upper bound on the largest possible number of vertices in any graph with this degree and diameter. Therefore, these graphs solve the degree
Moore_graph
System that regulates the formation of blocks on a blockchain
2018-04-09. Retrieved 2007-11-04. Tromp, John (2015). "Cuckoo Cycle: A Memory Bound Graph-Theoretic Proof-of-Work" (PDF). Financial Cryptography and Data Security
Proof_of_work
Basic concept of graph theory
mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that
Connectivity_(graph_theory)
Number denoting a graph's closeness to a tree
complexity analysis of graph algorithms. Many algorithms that are NP-hard for general graphs, become easier when the treewidth is bounded by a constant. The
Treewidth
Property of artificial neural networks
universal function approximation on bounded graphs and restricted universal function approximation on unbounded graphs, with an accompanying O ( | V | ⋅
Universal approximation theorem
Universal_approximation_theorem
Second-largest eigenvalue lower bound
In spectral graph theory, the Alon–Boppana bound provides a lower bound on the second-largest eigenvalue of the adjacency matrix of a d {\displaystyle
Alon–Boppana_bound
Subgraph with contracted edges
In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges and vertices and by contracting edges
Graph_minor
Regular graph with fewest possible nodes for its girth
of graph theory, a cage is a regular graph that has as few vertices as possible for its girth. Formally, an (r, g)-graph is defined to be a graph in which
Cage_(graph_theory)
Theorems connecting continuity to closure of graphs
analysis is whether a given linear operator is continuous (or bounded). The closed graph theorem gives one answer to that question. Let T : X → Y {\displaystyle
Closed graph theorem (functional analysis)
Closed_graph_theorem_(functional_analysis)
Algorithm for finding shortest paths
arbitrary directed graphs with unbounded non-negative weights. However, specialized cases (such as bounded/integer weights, directed acyclic graphs etc.) can be
Dijkstra's_algorithm
Finding the largest graph of given diameter and degree
size of G is bounded above by the Moore bound; for 1 < k and 2 < d, only the Petersen graph, the Hoffman-Singleton graph, and possibly graphs (not yet proven
Degree_diameter_problem
In mathematics, a topological graph is a representation of a graph in the plane, where the vertices of the graph are represented by distinct points and
Topological_graph
Class of mathematical games
exact upper bound for the game chromatic number of graphs in this class. This value is known for several standard graph classes, and bounded for some others:
Graph_coloring_game
Fewest edge crossings in drawing of a graph
graph theory, the crossing number cr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is
Crossing number (graph theory)
Crossing_number_(graph_theory)
American computer scientist and educator
logarithmic bound for approximating finite metrics by tree metrics. With Arora and Vazirani, Rao developed the expander-flow method for graph partitioning
Satish_B._Rao
Graph divided into two independent sets
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets
Bipartite_graph
Unrelated vertices in graphs
In graph theory, an independent set, stable set, coclique or anticlique is a set of vertices in a graph, no two of which are adjacent. That is, it is a
Independent set (graph theory)
Independent_set_(graph_theory)
Family of graphs whose shallow minors are sparse graphs
In graph theory, a family of graphs is said to have bounded expansion if all of its shallow minors are sparse graphs. Many natural families of sparse
Bounded_expansion
Set of edges without common vertices
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In
Matching_(graph_theory)
Mathematical problem
distance graph of the plane. Therefore, at least four colors are needed to color this graph and the plane containing it. An alternative lower bound in the
Hadwiger–Nelson_problem
any complete graph with fewer crossings than the number given by his upper bound? Universal point sets of subquadratic size for planar graphs Does there
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Graph linking pairs of comparable elements in a partial order
Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability
Comparability_graph
Trail in a graph that visits each edge once
In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices)
Eulerian_path
Subdivision of vertices into disjoint sets
In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges
Graph_partition
Algorithm used for pathfinding and graph traversal
outperformed by algorithms that can pre-process the graph to attain better performance, as well as by memory-bounded approaches; however, A* is still the best solution
A*_search_algorithm
Concept in graph theory
In graph theory, a strongly regular graph (SRG) is a regular graph G = (V, E) with v vertices and degree k such that for some given integers λ , μ ≥ 0
Strongly_regular_graph
Influence of local substructure of a graph on global properties
In essence, extremal graph theory studies how global properties of a graph influence local substructure. Results in extremal graph theory deal with quantitative
Extremal_graph_theory
Logical formulation of graph properties
the mathematical fields of graph theory and finite model theory, the logic of graphs deals with formal specifications of graph properties using sentences
Logic_of_graphs
Bijection between the vertex set of two graphs
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H f : V ( G ) → V ( H ) {\displaystyle f\colon V(G)\to
Graph_isomorphism
Graph representing edges of another graph
In the mathematical discipline of graph theory, the line graph of an undirected graph G is another graph L(G) that represents the adjacencies between edges
Line_graph
Graph without triples of adjacent vertices
area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently
Triangle-free_graph
Graph in which every two vertices are adjacent
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique
Complete_graph
Extremal graph theory bound on clique-free graph edges
In graph theory, Turán's theorem bounds the number of edges that can be included in an undirected graph that does not have a complete subgraph of a given
Turán's_theorem
Mapping a graph onto itself without changing edge-vertex connectivity
In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving
Graph_automorphism
Mathematical abstraction of level sets
the structure of a finite graph, then b 1 ( R f ) {\displaystyle b_{1}(R_{f})} is the cycle rank of this graph. An upper bound holds b 1 ( R f ) ≤ c o r
Reeb_graph
Graph of numbers differing by a square
Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic
Paley_graph
Type of graph in mathematical graph theory
discipline of graph theory, the (m,n)-lollipop graph is a special type of graph consisting of a complete graph (clique) on m vertices and a path graph on n vertices
Lollipop_graph
Any planar graph can be subdivided by removing a few vertices
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split
Planar_separator_theorem
Fewest cliques covering a graph's edges
In the mathematical field of graph theory, the intersection number of a graph G = ( V , E ) {\displaystyle G=(V,E)} is the smallest number of elements
Intersection number (graph theory)
Intersection_number_(graph_theory)
Measure of whether or not a graph has a "bottleneck"
Spectral graph theory Algebraic connectivity Cheeger bound Conductance Connectivity Expander graph Mohar 1989, pp. 274–291. Montenegro & Tetali 2006, pp
Cheeger constant (graph theory)
Cheeger_constant_(graph_theory)
Unproven generalization of the four-color theorem
in mathematics Does every graph with chromatic number k {\displaystyle k} have a k {\displaystyle k} -vertex complete graph as a minor? More unsolved
Hadwiger conjecture (graph theory)
Hadwiger_conjecture_(graph_theory)
Balanced complete multipartite graph
by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph. Via this theorem, similar bounds in extremal graph theory
Turán_graph
On tangency patterns of circles
surfaces of bounded genus. More generally, intersection graphs of interior-disjoint geometric objects are called tangency graphs or contact graphs. As a special
Circle_packing_theorem
Graph defined from a mathematical group
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract
Cayley_graph
zig-zag product to efficiently construct expander graphs. Jesper Jansson. Deterministic Space-Bounded Graph Connectivity Algorithms. Manuscript. 1998. Harry
Symmetric_Turing_machine
Graph with at most one crossing per edge
In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing
1-planar_graph
Undirected unit-distance graph requiring four colors
a unit distance graph that requires four colors in any graph coloring. Thus, like the simpler Moser spindle, it provides a lower bound for the Hadwiger–Nelson
Golomb_graph
NP-hard problem in combinatorial optimization
improve the lower bound, a better way of creating an Eulerian graph is needed. By the triangle inequality, the best Eulerian graph must have the same
Travelling_salesman_problem
In graph theory, a χ {\displaystyle \chi } -bounded (using the Greek letter chi) family F {\displaystyle {\mathcal {F}}} of graphs is one for which there
Chi-bounded
Visualization of node-link graphs
number of degree-4 graphs is bounded. There are many different graph layout strategies: In force-based layout systems, the graph drawing software modifies
Graph_drawing
discovered, and thus finding a larger graph that is closer in order (in terms of the size of the vertex set) to the Moore bound is considered an open problem
Table of the largest known graphs of a given diameter and maximal degree
Table_of_the_largest_known_graphs_of_a_given_diameter_and_maximal_degree
Real function with finite total variation
function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function
Bounded_variation
Heuristic test for graph isomorphism
them, and architectures attaining this bound have exactly the same power to distinguish non-isomorphic graphs as 1-WL. Higher-dimensional (k-WL) tests
Weisfeiler Leman graph isomorphism test
Weisfeiler_Leman_graph_isomorphism_test
British Othello player (born 1963)
combinatorics. Cited results included the proof, with Reinhard Diestel, of the bounded graph conjecture of Rudolf Halin. Leader in an interview in 2016 stated that
Imre_Leader
Graph partition into regular subgraphs
In extremal graph theory, Szemerédi's regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between
Szemerédi_regularity_lemma
Intersection graph of a chord diagram
In graph theory, a circle graph is the intersection graph of a chord diagram. That is, it is an undirected graph whose vertices can be associated with
Circle_graph
Number of vertices with unambiguous distances
interval graphs, and more generally to graphs of bounded tree-length, such as chordal graphs, permutation graphs or asteroidal-triple-free graphs. Deciding
Metric dimension (graph theory)
Metric_dimension_(graph_theory)
Assignment of colors to edges of a graph
In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color
Edge_coloring
Measure of graph complexity
i to label j (denoted by ρ(i,j)) Graphs of bounded clique-width include the cographs and distance-hereditary graphs. Although it is NP-hard to compute
Clique-width
Second-smallest eigenvalue of a graph Laplacian
negative for general directed graphs, even if G is a connected graph. Furthermore, the value of the algebraic connectivity is bounded above by the traditional
Algebraic_connectivity
Exponentially decreasing bounds on tail distributions of random variables
ISBN 978-3-540-42493-2.; lemma 6.1 See graphs of: the bound as a function of r when k changes and the bound as a function of k when r changes. Mulzer
Chernoff_bound
Graph of short distances in another graph
vertices than the original graph. If a graph has diameter d, then its d-th power is the complete graph. If a graph family has bounded clique-width, then so
Graph_power
Measure of the structural complexity of a software program
Cyclomatic complexity is computed using the control-flow graph of the program. The nodes of the graph correspond to indivisible groups of commands of a program
Cyclomatic_complexity
Maximum number of colors obtainable by a greedy graph coloring algorithm
chordal graphs and claw-free graphs, and also (using general results on subgraph isomorphism in sparse graphs to search for atoms) for graphs of bounded expansion
Grundy_number
Theorem in extremal graph theory
extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an H-free graph for a
Erdős–Stone_theorem
Minimum spanning forest algorithm that greedily adds edges
algorithm finds a minimum spanning forest of an undirected edge-weighted graph. If the graph is connected, it finds a minimum spanning tree. It is a greedy algorithm
Kruskal's_algorithm
Mathematical graph relating to chess
mathematics, a queen's graph is an undirected graph that represents all legal moves of the queen—a chess piece—on a chessboard. In the graph, each vertex represents
Queen's_graph
Longest distance between tree vertices
In graph theory, the triameter is a metric invariant that generalizes the concept of a graph's diameter. It is defined as the maximum sum of pairwise
Triameter_(graph_theory)
Graph with almost the max amount of edges
dense. The classes of graphs with bounded degeneracy and of nowhere dense graphs are both included in the biclique-free graphs, graph families that exclude
Dense_graph
Upper bound on a graph's Shannon capacity
In graph theory, the Lovász number of a graph is a real number that is an upper bound on the Shannon capacity of the graph. It is also known as Lovász
Lovász_number
Partition of a graph whose components are reachable from all vertices
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly
Strongly_connected_component
Theorem on graph coloring on surfaces
In graph theory, the Heawood conjecture or Ringel–Youngs theorem gives a lower bound for the number of colors that are necessary for graph coloring on
Heawood_conjecture
Query language for property graphs
results are produced by matching graph patterns and returning variables bound to nodes, edges, or paths. Current graph database products and projects often
Graph_Query_Language
Graph representing faces of another graph
mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each
Dual_graph
Measurement of graph sparsity
In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree at most k {\displaystyle k} . That
Degeneracy_(graph_theory)
Algorithmically defined graph
In the study of graph algorithms, an implicit graph representation (or more simply implicit graph) is a graph whose vertices or edges are not represented
Implicit_graph
On minimizing crossings in bicliques
bipartite graph requires a number of crossings that is (for sufficiently large graphs) at least 83% of the number given by the Zarankiewicz bound. Closing
Turán's_brick_factory_problem
Representation of a graph as a path graph "thickened" by some amount
minor-closed graph families have bounded pathwidth. Pathwidth, and graphs of bounded pathwidth, also have applications in VLSI design, graph drawing, and
Pathwidth
BOUND GRAPH
BOUND GRAPH
Surname or Lastname
English
English : variant of Bond
Boy/Male
American, Australian, British, Christian, English, German, Indian
Tied to the Land; Tiller of the Soil; Farmer
Male
English
Farmer
Girl/Female
Australian, Danish, Finnish, German, Hebrew, Japanese, Netherlands, Scandinavian
Blind; Bound
Surname or Lastname
English
English : status name for a peasant farmer or husbandman, Middle English bonde (Old English bonda, bunda, reinforced by Old Norse bóndi). The Old Norse word was also in use as a personal name, and this has given rise to other English and Scandinavian surnames alongside those originating as status names. The status of the peasant farmer fluctuated considerably during the Middle Ages; moreover, the underlying Germanic word is of disputed origin and meaning. Among Germanic peoples who settled to an agricultural life, the term came to signify a farmer holding lands from, and bound by loyalty to, a lord; from this developed the sense of a free landholder as opposed to a serf. In England after the Norman Conquest the word sank in status and became associated with the notion of bound servitude.Swedish : variant of Bonde.
Girl/Female
Assamese, Hindu, Indian, Marathi
Bound
Surname or Lastname
English (chiefly West Midlands)
English (chiefly West Midlands) : nickname for a plump person, from Middle English, Old French rond, rund ‘fat’, ‘round’ (Latin rotundus).
Biblical
bound; limit
Boy/Male
English
Tied to the land.
Surname or Lastname
English
English : presumably a variant of Mount.
Girl/Female
Biblical
Bound, limit.
Girl/Female
Hindu, Indian, Tamil
Drop
Surname or Lastname
English
English : variant of Bond.
Girl/Female
Indian
A bond, One who glues together, Is bound, Preserve
Surname or Lastname
English (also found in Ireland)
English (also found in Ireland) : from a pet form of Lamb 1 and 2.
Boy/Male
Indian, Marathi
Raindrops
Surname or Lastname
English
English : from Middle English p(o)und ‘enclosure (especially for confining animals)’; a topographic name for someone who lived near an enclosure in which animals were kept, or a metonymic occupational name for an official responsible for rounding up stray animals and placing them in a pound.Probably a translation of German Pfund or the North German cognate Pund.
Surname or Lastname
English (found mainly in Wales)
English (found mainly in Wales) : variant of Glasscock 2.
Surname or Lastname
English
English : patronymic from Bond.
Girl/Female
Tamil
Bandini | பநà¯à®¤à¯€à®¨à¯€Â
A bond, One who glues together, Is bound, Preserve
BOUND GRAPH
BOUND GRAPH
Boy/Male
Indian
Treasure
Boy/Male
Welsh
god of the seas'.
Boy/Male
American, Anglo, Australian, British, English, French
From the Riverbank Enclosure
Boy/Male
Greek
Gracious gift.
Girl/Female
Hindu, Indian, Marathi
Lovable; Lord Shiva / Ganesha
Girl/Female
Hindu
Lustrous or bright or radiant or intelligent, Brave, Powerful
Girl/Female
American, British, English
Joyous; Playful
Boy/Male
Arabic, Muslim
A Cast of Afghans; Maker
Boy/Male
American, Australian, British, Chinese, Christian, Danish, English
Dear Friend
Boy/Male
Arabic
One who Praises Allah (Dikr)
BOUND GRAPH
BOUND GRAPH
BOUND GRAPH
BOUND GRAPH
BOUND GRAPH
a.
Outspoken; plain and direct; unreserved; unqualified; not mincing; as, a round answer; a round oath.
n.
That which goes round a whole circle or company; as, a round of applause.
n.
Anything round, as a circle, a globe, a ring. "The golden round" [the crown].
superl.
Whole; unbroken; unharmed; free from flaw, defect, or decay; perfect of the kind; as, sound timber; sound fruit; a sound tooth; a sound ship.
n.
The occasion of sound; the impulse or vibration which would occasion sound to a percipient if present with unimpaired; hence, the theory of vibrations in elastic media such cause sound; as, a treatise on sound.
v. t.
To make to bound or leap; as, to bound a horse.
n.
Rebound; as, the bound of a ball.
v. t.
To order, direct, indicate, or proclain by a sound, or sounds; to give a signal for by a certain sound; as, to sound a retreat; to sound a parley.
superl.
Founded in truth or right; supported by justice; not to be overthrown on refuted; not fallacious; as, sound argument or reasoning; a sound objection; sound doctrine; sound principles.
p. p. & a.
Constrained or compelled; destined; certain; -- followed by the infinitive; as, he is bound to succeed; he is bound to fail.
p. p. & a.
Resolved; as, I am bound to do it.
p. p. & a.
Inclosed in a binding or cover; as, a bound volume.
n.
The state of being bound; imprisonment; captivity, restraint.
pl.
of Pound
superl.
Healthy; not diseased; not being in a morbid state; -- said of body or mind; as, a sound body; a sound constitution; a sound understanding.
v. i.
To be conveyed in sound; to be spread or published; to convey intelligence by sound.
v. t.
To cause to rebound; to throw so that it will rebound; as, to bound a ball on the floor.
v. t.
To go round wholly or in part; to go about (a corner or point); as, to round a corner; to round Cape Horn.
a.
Uttered or emitted with a full tone; as, a round voice; a round note.
v. t.
To name the boundaries of; as, to bound France.