91Ó°ÊÓ

Graph is the pictorial representation that is the combinations of nodes and edges.Degree is the total number of edges a node is connected.

Consider a graph$G$ which has at least one edge, without having any loops or cycles. In graph$G$ , prove that there are at least two nodes with degree 1.

In a graph $G$, consider the node $V_1$at which only one edge $E_1$is incident.

$\mathrm{deg}\left(V_1\right)=1$

Consider that the other end of the $E_1$is at $V_2$. If there exists no other edges which is incident at $V_2$.

$\mathrm{deg}\left(V_2\right)=1$

Let$E_2$ be the edge which is incident at $V_2$. Proceeding like this, node$V_k$ will be obtained that has the degree value 1 and it is equal to the degree of$V_1$ node.

Thus, the graph$G$ has at least two nodes of degree 1.

Therefore, it has been proved that every graph with two or more nodes contains two nodes that have equal degrees.