# Edge vs. Vertex — What's the Difference?

By Fiza Rafique & Urooj Arif — Updated on May 9, 2024
An edge in a graph theory represents a connection between two vertices, while a vertex is a point where edges meet or terminate.

## Key Differences

In graph theory, an edge is a line that connects two vertices, suggesting a relationship or pathway between them. A vertex (or node), however, is one of the fundamental units that make up the structure of a graph, representing endpoints or intersections of edges.
Edges are crucial for defining the connectivity between vertices, enabling the graph to illustrate networks such as social connections, computer networks, or maps. In contrast, vertices serve as critical points of data, decisions, or locations within the graph.
The role of an edge is to express the type and strength of the relationship between vertices. For example, in weighted graphs, edges have values that represent the cost or distance between vertices. Meanwhile, vertices might be used to represent entities such as cities in a map, where each city is a vertex.
Graphically, an edge is depicted as a line or arc that may be directed (indicating one-way interaction) or undirected. Vertices are usually shown as dots or circles, often labeled with letters or numbers to provide information or to identify them.
In algorithms, the efficiency often depends on the number of vertices and edges in a graph. Operations such as traversing or searching a graph involve moving through edges to reach various vertices, highlighting their interdependent nature.

## Comparison Chart

### Definition

Connects two vertices in a graph
A point in a graph where edges meet

### Role in Graphs

Indicates relationships and pathways
Acts as a fundamental unit, a data point

### Graphical Representation

Depicted as lines or arcs between dots
Represented as dots or circles in a graph

### Importance in Algorithms

Used to traverse between points in a graph
Served as critical points for data or decisions

### Example in Practical Use

Cities or intersections in a transportation network

## Compare with Definitions

#### Edge

A line connecting two vertices in a graph.
In a friendship network graph, an edge represents a friendship link between two people.

#### Vertex

Represents entities or locations in models.
Each vertex in a city traffic map might represent an intersection or endpoint.

#### Edge

Often carries weight in weighted graphs.
An edge in a network graph might represent the distance or cost between nodes.

#### Vertex

A point where edges converge in a graph.
In a computer network, each computer is a vertex connected by network links (edges).

#### Edge

Can be directed or undirected.
A directed edge in a graph indicates a one-way street between two points.

#### Vertex

Labeled for identification in complex graphs.
Vertices in a molecular structure are labeled with element symbols.

#### Edge

Represents relationships in abstract models.
An edge in a project dependency graph shows the prerequisite relationships between tasks.

#### Vertex

Used to represent decision or data points in algorithms.
Each vertex in a decision tree represents a decision or classification point.

#### Edge

Integral for defining graph structure.
Edges determine the connectivity pattern in a neural network.

#### Vertex

Can be entry or exit points in pathfinding.
Vertices in a maze are critical points where decisions are made.

#### Edge

The outside limit of an object, area, or surface
She perched on the edge of a desk
A willow tree at the water's edge

#### Vertex

The highest point; the top or apex.

#### Edge

The sharpened side of the blade of a cutting implement or weapon
A knife with a razor-sharp edge

#### Vertex

Each angular point of a polygon, polyhedron, or other figure.

#### Edge

A quality or factor which gives superiority over close rivals
His cars have the edge over his rivals'

#### Vertex

The highest point; the apex or summit
The vertex of a mountain.

#### Edge

Provide with a border or edge
The pool is edged with paving

#### Vertex

The highest point of the skull.

#### Edge

Move or cause to move gradually or furtively in a particular direction
Hazel quietly edged him away from the others
She tried to edge away from him

#### Edge

Give an intense or sharp quality to
The bitterness that edged her voice

#### Vertex

In astrology, the highest point reached in the apparent motion of a celestial body.

#### Edge

Strike (the ball) with the edge of the bat; strike a ball delivered by (the bowler) with the edge of the bat
Haynes edged to slip
He edged a ball into his pad

#### Vertex

The point at which the sides of an angle intersect.

#### Edge

Ski with one's weight on the edges of one's skis
You will be edging early, controlling a parallel turn

#### Vertex

The point on a triangle or pyramid opposite to and farthest away from its base.

#### Edge

A thin, sharpened side, as of the blade of a cutting instrument.

#### Vertex

A point on a polyhedron common to three or more sides.

#### Edge

The degree of sharpness of a cutting blade.

#### Vertex

A point of maximal curvature on a parabola or hyperbola.

#### Edge

A penetrating, incisive quality
"His simplicity sets off the satire, and gives it a finer edge" (William Hazlitt).

#### Vertex

The highest point, top or apex of something.

#### Edge

A slight but noticeable sharpness, harshness, or discomforting quality
His voice had an edge to it.

#### Vertex

(anatomy) The highest surface on the skull; the crown of the head.

#### Edge

Keenness, as of desire or enjoyment; zest
The brisk walk gave an edge to my appetite.

#### Vertex

(geometry) An angular point of a polygon, polyhedron or higher order polytope.

#### Edge

The line or area farthest away from the middle
Lifted the carpet's edge.

#### Vertex

The common point of the two rays that form an angle.

#### Edge

The line of intersection of two surfaces
The edge of a brick.

#### Vertex

The point at which an axis meets a curve or surface.

#### Edge

A rim or brink
The edge of a cliff.

#### Vertex

(mathematics) A point on the curve with a local minimum or maximum of curvature.

#### Edge

The point at which something is likely to begin
On the edge of war.

#### Vertex

(graph theory) One of the elements of a graph joined or not by edges to other vertices.

#### Edge

A margin of superiority; an advantage
A slight edge over the opposition.

#### Vertex

(computer graphics) A point in 3D space, usually given in terms of its Cartesian coordinates.

#### Edge

To give an edge to (a blade); sharpen.

#### Vertex

(optics) The point where the surface of a lens crosses the optical axis.

#### Edge

To tilt (a ski or both skis) in such a way that an edge or both edges bite into the snow.

#### Vertex

(particle physics) An interaction point.

#### Edge

To put a border or edge on
Edged the quilt with embroidery.

#### Vertex

(astrology) The point where the prime vertical meets the ecliptic in the western hemisphere of a natal chart.

#### Edge

To act as or be an edge of
Bushes that edged the garden path.

#### Vertex

(typography) A sharp downward point opposite a crotch, as in the letters "V" and "W" but not "Y".

#### Edge

The dog edged the ball with its nose.

#### Vertex

A turning point; the principal or highest point; top; summit; crown; apex.

#### Edge

To trim or shape the edge of
Edge a lawn.

#### Vertex

The top, or crown, of the head.

#### Edge

To surpass or beat by a small margin. Often used with out
The runner edged her opponent out at the last moment.

#### Vertex

The zenith, or the point of the heavens directly overhead.

#### Edge

The child edged toward the door.

#### Vertex

The point in any figure opposite to, and farthest from, the base; the terminating point of some particular line or lines in a figure or a curve; the top, or the point opposite the base.

#### Edge

The boundary line of a surface.

#### Vertex

The point of intersection of lines or the point opposite the base of a figure

#### Edge

(geometry) A one-dimensional face of a polytope. In particular, the joining line between two vertices of a polygon; the place where two faces of a polyhedron meet.

#### Vertex

The highest point (of something);
At the peak of the pyramid

#### Edge

I have the edge on him.

#### Edge

The thin cutting side of the blade of an instrument, such as an ax, knife, sword, or scythe; that which cuts as an edge does, or wounds deeply, etc.

#### Edge

A sharp terminating border; a margin; a brink; an extreme verge.
The cup is right on the edge of the table.
He is standing on the edge of a precipice.

#### Edge

Sharpness; readiness or fitness to cut; keenness; intenseness of desire.

#### Edge

The border or part adjacent to the line of division; the beginning or early part (of a period of time)
In the edge of evening

#### Edge

(cricket) A shot where the ball comes off the edge of the bat, often unintentionally.

#### Edge

(graph theory) A connected pair of vertices in a graph.

#### Edge

The point of data production in an organization (the focus of edge computing), as opposed to the cloud.

#### Edge

(transitive) To move an object slowly and carefully in a particular direction.
He edged the book across the table.
The muggers edged her into an alley and demanded money.

#### Edge

(intransitive) To move slowly and carefully in a particular direction.
He edged away from her.

#### Edge

(usually in the form 'just edge') To win by a small margin.

#### Edge

To hit the ball with an edge of the bat, causing a fine deflection.

#### Edge

(transitive) To trim the margin of a lawn where the grass meets the sidewalk, usually with an electric or gas-powered lawn edger.

#### Edge

(transitive) To furnish with an edge; to construct an edging.

#### Edge

To furnish with an edge, as a tool or weapon; to sharpen.

#### Edge

(figurative) To make sharp or keen; to incite; to exasperate; to goad; to urge or egg on.

#### Edge

To delay one's orgasm so as to remain almost at the point of orgasm.

#### Edge

The thin cutting side of the blade of an instrument; as, the edge of an ax, knife, sword, or scythe.
He which hath the sharp sword with two edges.
Slander,Whose edge is sharper than the sword.

#### Edge

Any sharp terminating border; a margin; a brink; extreme verge; as, the edge of a table, a precipice.
Upon the edge of yonder coppice.
In worst extremes, and on the perilous edgeOf battle.
Pursue even to the very edge of destruction.

#### Edge

Sharpness; readiness or fitness to cut; keenness; intenseness of desire.
The full edge of our indignation.
Death and persecution lose all the ill that they can have, if we do not set an edge upon them by our fears and by our vices.

#### Edge

The border or part adjacent to the line of division; the beginning or early part; as, in the edge of evening.

#### Edge

To furnish with an edge as a tool or weapon; to sharpen.
To edge her champion's sword.

#### Edge

To shape or dress the edge of, as with a tool.

#### Edge

To furnish with a fringe or border; as, to edge a dress; to edge a garden with box.
Hills whose tops were edged with groves.

#### Edge

To make sharp or keen, figuratively; to incite; to exasperate; to goad; to urge or egg on.
By such reasonings, the simple were blinded, and the malicious edged.

#### Edge

To move by little and little or cautiously, as by pressing forward edgewise; as, edging their chairs forwards.

#### Edge

To move sideways; to move gradually; as, edge along this way.

#### Edge

To sail close to the wind.
I must edge up on a point of wind.

#### Edge

The boundary of a surface

#### Edge

A sharp side formed by the intersection of two surfaces of an object;
He rounded the edges of the box

#### Edge

A line determining the limits of an area

#### Edge

The attribute of urgency;
His voice had an edge to it

#### Edge

He had an edge on the competition

#### Edge

A strip near the boundary of an object;
He jotted a note on the margin of the page

#### Edge

Advance slowly, as if by inches;
He edged towards the car

#### Edge

Provide with a border or edge;
Edge the tablecloth with embroidery

#### Edge

Lie adjacent to another or share a boundary;
England marches with Scotland

#### Edge

Provide with an edge;

## Common Curiosities

#### What is a vertex in graph theory?

A point in a graph where two or more edges meet, acting as a fundamental unit.

#### Are edges more important than vertices in a graph?

Both are crucial for different reasons; edges define relationships, while vertices serve as points of connection or data.

#### Can a graph exist without edges?

Yes, such a graph would simply have isolated vertices with no connections.

#### Why are vertices important in algorithms?

They often represent critical data points or decision nodes in computational processes.

#### Can a graph exist without vertices?

No, a graph must have at least one vertex; without vertices, there’s nothing to connect.

#### What is an edge in graph theory?

It’s a line connecting two vertices, representing a relationship or pathway.

#### How many edges can connect to a single vertex?

It varies; a vertex can have as few edges as none or multiple, depending on the graph structure.

#### How are edges represented in a directed graph?

As arrows pointing from one vertex to another, indicating the direction of the relationship.

#### What does a weighted edge represent?

It typically represents a value like cost, distance, or capacity between vertices in a graph.

#### What’s an example of a real-world application of vertices and edges?

Traffic management systems use vertices to represent intersections and edges to represent roads, helping in route optimization and traffic flow analysis.

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