Find the number of vertices the number of edges and the degr

Find the number of vertices, the number of edges and the degree of each vertex in the undirected graph. Identify all isolated and pendant vertices. Find the sum of the degrees of the vertices and verify the Handshaking Theorem. State explicitly what this is equal to in terms of the edges.

Solution

all the corner points in the above graph cmprise of the vertices

so the number of vertices are = 5 {a , b , c , d , e}

all the lines joining two vertices are the edges , the rings at vertices a and c has the same start and the end point that is a and c respecitely , they are edges as well.

=> the number of vertices are = 13

the degree of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice.

so deg(a) = 6 , deg(b) = 6 , deg(c) = 6 , deg(d) = 5 , deg(e) = 3

if there is a vertex with degree 0 that is its not connected with some other vertex or with itself then the vertex is called an isolated vertex

there is no isolated vertex in the given graph

A pendant vertex is a vertex whose degree is 1

there are no pendant vertex in the given graph

In any graph, the sum of the degrees of all vertices is equal to twice the number of edges.

=> sun of degree of vertices = (6+6+6+5+3)*2 = 26

Handshaking Theorem: The sum of the degrees of the vertices in a graph is twice the number of edges.

which is true for our graph , sum = 26.

we could see that sum of the degree of the vertices = 2*number of the edges

=> 26/2 = 13 edges