# PathGraph

PathGraph[{v1,v2,}]

yields a path with vertices vi and edges between vi and vi+1 .

PathGraph[{e1,e2,}]

yields a path with edges ej.

PathGraph[{v1,v2,},{e1,e2,}]

yields a path with vertices vi and edges ej.

PathGraph[{,wi[vi,],},{,wj[ej,],}]

yields a path with vertex and edge properties defined by the symbolic wrappers wk.

PathGraph[{vivj,}]

uses rules vi->vj to specify a path.

# Details and Options • PathGraph generates a Graph object.
• PathGraph supports the same vertices, edges, wrappers, and options as Graph.
• An undirected path graph is a connected graph where each vertex has at most degree two.
• A directed path graph is a connected graph where each vertex has at most in-degree one and at most out-degree one.
• PathGraph can only represent self-avoiding paths, except for cycles.

# Examples

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## Basic Examples(2)

A path constructed from a list of vertices:

A list of edges:

## Scope(6)

### Connectivity(6)

Create an undirected graph using characters, entering the character as ue :

Create a directed graph using characters, entering the character as de :

Create a directed graph from a list of rules:

Create an undirected graph from a list of rules:

Use VertexList and EdgeList to get vertices and edges:

The ordering for edges is the order in which they were entered:

The ordering for vertices is the order in which they entered in the edges:

Use an explicit vertex list to control the ordering used by VertexList:

The input vertex list controls the resulting vertex order:

Any expression can be used as vertices:

## Options(82)

### AnnotationRules(3)

Specify an annotation for vertices:

Edges:

Graph itself:

### DirectedEdges(2)

By default, a directed path is generated when giving a list of rules:

Use to interpret rules as undirected edges:

Use DirectedEdge or UndirectedEdge to directly specify whether a graph is directed or not:

### EdgeLabels(7)

Label the edge 12:

Label all edges individually:

Use any expression as a label:

Use Placed with symbolic locations to control label placement along an edge:

Use explicit coordinates to place labels:

Vary positions within the label:

Place multiple labels using Placed in a wrapper:

Any number of labels can be used:

Place multiple labels using EdgeLabels:

Use automatic labeling by values through Tooltip and StatusArea:

### EdgeShapeFunction(6)

Get a list of built-in settings for EdgeShapeFunction:

Undirected edges including the basic line:

Lines with different glyphs on the edges:

Directed edges including solid arrows:

Line arrows:

Open arrows:

Specify an edge function for an individual edge:

Combine with a different default edge function:

Draw edges by running a program:

EdgeShapeFunction can be combined with EdgeStyle:

EdgeShapeFunction has higher priority than EdgeStyle:

### EdgeStyle(2)

Style all edges:

Style individual edges:

### EdgeWeight(2)

Specify a weight for all edges:

Use any numeric expression as a weight:

### GraphHighlight(3)

Highlight the vertex 1:

Highlight the edge 23:

Highlight vertices and edges:

### GraphHighlightStyle(2)

Get a list of built-in settings for GraphHighlightStyle:

Use built-in settings for GraphHighlightStyle:

### GraphLayout(5)

By default, the layout is chosen automatically:

Specify layouts on special curves:

Specify layouts that satisfy optimality criteria:

VertexCoordinates overrides GraphLayout coordinates:

Use AbsoluteOptions to extract VertexCoordinates computed using a layout algorithm:

### PlotTheme(4)

#### Base Themes(2)

Use a common base theme:

Use a monochrome theme:

#### Feature Themes(2)

Use a large graph theme:

Use a classic diagram theme:

### VertexCoordinates(3)

By default, any vertex coordinates are computed automatically:

Extract the resulting vertex coordinates using AbsoluteOptions:

Specify a layout function along an ellipse:

Use it to generate vertex coordinates for a graph:

VertexCoordinates has higher priority than GraphLayout:

### VertexLabels(13)

Use vertex names as labels:

Label individual vertices:

Label all vertices:

Use any expression as a label:

Use Placed with symbolic locations to control label placement, including outside positions:

Symbolic outside corner positions:

Symbolic inside positions:

Symbolic inside corner positions:

Use explicit coordinates to place the center of labels:

Place all labels at the upper-right corner of the vertex and vary the coordinates within the label:

Place multiple labels using Placed in a wrapper:

Any number of labels can be used:

Place multiple labels using VertexLabels:

Use the argument to Placed to control formatting including Tooltip:

Or StatusArea:

Use more elaborate formatting functions:

### VertexShape(5)

Use any Graphics, Image, or Graphics3D as a vertex shape:

Specify vertex shapes for individual vertices:

VertexShape can be combined with VertexSize:

VertexShape is not affected by VertexStyle:

VertexShapeFunction has higher priority than VertexShape:

### VertexShapeFunction(10)

Get a list of built-in collections for VertexShapeFunction:

Use built-in settings for VertexShapeFunction in the "Basic" collection:

Simple basic shapes:

Common basic shapes:

Use built-in settings for VertexShapeFunction in the "Rounded" collection:

Use built-in settings for VertexShapeFunction in the "Concave" collection:

Draw individual vertices:

Combine with a default vertex function:

Draw vertices using a predefined graphic:

Draw vertices by running a program:

VertexShapeFunction can be combined with VertexStyle:

VertexShapeFunction has higher priority than VertexStyle:

VertexShapeFunction can be combined with VertexSize:

VertexShapeFunction has higher priority than VertexShape:

### VertexSize(8)

By default, the size of vertices is computed automatically:

Specify the size of all vertices using symbolic vertex size:

Use a fraction of the minimum distance between vertex coordinates:

Use a fraction of the overall diagonal for all vertex coordinates:

Specify size in both the and directions:

Specify a size for individual vertices:

VertexSize can be combined with VertexShapeFunction:

VertexSize can be combined with VertexShape:

### VertexStyle(5)

Style all vertices:

Style individual vertices:

VertexShapeFunction can be combined with VertexStyle:

VertexShapeFunction has higher priority than VertexStyle:

VertexStyle can be combined with BaseStyle:

VertexStyle has higher priority than BaseStyle:

VertexShape is not affected by VertexStyle:

### VertexWeight(2)

Set the weight for all vertices:

Use any numeric expression as a weight:

## Applications(6)

The GraphCenter of path graphs:

The GraphPeriphery:

Highlight the vertex eccentricity path:

The GraphDiameter:

Highlight the diameter path:

Visualize different centralities for PathGraph:

Highlight the closeness centrality:

Highlight the eigenvector centrality:

## Properties & Relations(10)

Use VertexCount and EdgeCount to count vertices and edges:

Use VertexList and EdgeList to enumerate vertices and edges in standard order:

Edges and vertices are given in the order they are input:

Rows and columns of the adjacency matrix follow the order given by VertexList:

Compute the IncidenceMatrix from a graph:

The row ordering is given by VertexList and column ordering is given by EdgeList:

A path graph is a loop-free graph:

A path graph that starts and ends in the same vertex is a cycle graph:

A path graph is connected and each vertex has at most degree 2:

A path graph with no repeated vertices is a tree:

A path graph with no repeated vertices is acyclic:

The line graph of a path is isomorphic to :