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<h1 id="94-summary">9.4 &nbsp; Summary<a class="headerlink" href="#94-summary" title="Permanent link">&para;</a></h1>
<h3 id="1-key-takeaways">1. &nbsp; Key takeaways<a class="headerlink" href="#1-key-takeaways" title="Permanent link">&para;</a></h3>
<ul>
<li>A graph consists of vertices and edges and can be represented using a vertex set and an edge set.</li>
<li>Network relationships (graphs) allow more freedom than linear relationships (linked lists) or divide-and-conquer relationships (trees), making them more complex.</li>
<li>Edges in a directed graph have a direction; all vertices in a connected graph are reachable; and each edge in a weighted graph has a weight.</li>
<li>An adjacency matrix represents a graph with a matrix. Each row (or column) represents a vertex, and entries represent edges, with <span class="arithmatex">\(1\)</span> or <span class="arithmatex">\(0\)</span> indicating whether an edge exists between two vertices. Adding, removing, looking up, and modifying edges are efficient, but the matrix uses substantial space.</li>
<li>An adjacency list represents a graph with multiple linked lists. The <span class="arithmatex">\(i\)</span>-th linked list corresponds to vertex <span class="arithmatex">\(i\)</span> and stores all of its adjacent vertices. It uses less space than an adjacency matrix, but finding an edge requires traversing a linked list, making lookups slower.</li>
<li>A long linked list in an adjacency list can be replaced with a red-black tree or a hash table to speed up lookups.</li>
<li>In terms of algorithm design, an adjacency matrix trades space for time, while an adjacency list trades time for space.</li>
<li>Graphs can model many real-world systems, such as social networks and subway networks.</li>
<li>A tree is a special case of a graph, and tree traversal is a special case of graph traversal.</li>
<li>Breadth-first graph traversal searches from near to far, expanding layer by layer. It is usually implemented with a queue.</li>
<li>Depth-first graph traversal follows a path as far as possible and backtracks when it can go no farther. It is often implemented with recursion.</li>
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<h3 id="2-q-a">2. &nbsp; Q &amp; A<a class="headerlink" href="#2-q-a" title="Permanent link">&para;</a></h3>
<p><strong>Q</strong>: Is a path defined as a sequence of vertices or a sequence of edges?</p>
<p>Wikipedia's language editions give different definitions: the English edition defines a path as a sequence of edges, while the Chinese edition defines it as a sequence of vertices. The English wording is: In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices.</p>
<p>In this book, a path is treated as a sequence of edges rather than vertices. This is because multiple edges can connect the same two vertices, in which case each edge corresponds to a path.</p>
<p><strong>Q</strong>: Can some vertices in a disconnected graph remain unvisited?</p>
<p>In a disconnected graph, at least one vertex cannot be reached from a given starting vertex. Traversing the entire graph requires multiple starting points so that all connected components are visited.</p>
<p><strong>Q</strong>: Must the adjacent vertices in an adjacency list appear in a particular order?</p>
<p>They may appear in any order. In practice, however, it can be useful to sort them by a specific rule, such as the order in which vertices were added or their values. This helps us quickly find a vertex with a particular extreme value.</p>
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