# isConnectedGraph -- determines if a graph is connected

## Synopsis

• Usage:
b = isConnectedGraph G
• Inputs:
• G, ,
• Outputs:
• b, , true if G is connected, false otherwise

## Description

This function checks if the given graph G is connected. A graph is said to be connected if it has exactly one connected component.

Isolated vertices form their own connected components and will cause this method return false. This is in contrast to isConnected in which isolated vertices are not in any connected components. See the Connected Components Tutorial for more information.

 i1 : S = QQ[a..e]; i2 : G = graph {a*b,b*c,c*d,d*e,a*e} -- the 5-cycle (connected) o2 = Graph{edges => {{a, b}, {b, c}, {c, d}, {a, e}, {d, e}}} ring => S vertices => {a, b, c, d, e} o2 : Graph i3 : H = graph {a*b,b*c,c*a,d*e} -- a 3-cycle and a disjoint edge (not connected) o3 = Graph{edges => {{a, b}, {a, c}, {b, c}, {d, e}}} ring => S vertices => {a, b, c, d, e} o3 : Graph i4 : isConnectedGraph G o4 = true i5 : isConnectedGraph H o5 = false

In the following example, the graph G has the isolated vertex e. As d forms its own connected component, this graph is not connected.

 i6 : S = QQ[a..e]; i7 : G = graph {a*b,b*c,c*d,a*d} -- 4-cycle with isolated vertex (not connected) o7 = Graph{edges => {{a, b}, {b, c}, {a, d}, {c, d}}} ring => S vertices => {a, b, c, d, e} o7 : Graph i8 : isolatedVertices G o8 = {e} o8 : List i9 : isConnectedGraph G o9 = false