Word Count : 981

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In graph theory, the relationship between vertices and edges is fundamental. Let $G = (V, E)$ be a simple graph with vertex set $V$ and edge set $E$. The degree of a vertex $v \in V$, denoted $deg(v)$, is the number of edges incident to $v$. To show that the sum of the degrees of all vertices of $G$ equals twice the number of edges, consider each edge in $E$. Every edge connects two vertices, and thus contributes 1 to the degree of each endpoint. Summing over all vertices counts each edge exactly twice, once for each endpoint. Formally, $\sum_{v \in V} deg(v) = 2|E|$. This result is known as the Handshaking Lemma and holds for all simple, undirected graphs, including multigraphs if loops are counted twice. Next, we consider the principle of induction to prove that if $W$ is a $u-v$ walk joining distinct vertices $u$ and $v$, then there exists a path connecting $u$ and $v$ contained within $W$. A walk __________ __________ ____ _____ _______.
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