Question

Construct two different adjacency matrices and two different adjacency lists for $C_4$.

Short answer

Two adjacency matrices and lists are constructed for $C_4$.

Step-by-step solution

Define the graph

The cycle graph $C_4$ consists of 4 vertices connected in a closed loop. Let's denote these vertices as $V_1,V_2,V_3,$ and $V_4$. In $C_4$, the edges are $(V_1,V_2),(V_2,V_3),(V_3,V_4),(V_4,V_1)$.

Construct the first adjacency matrix

In the adjacency matrix for $C_4$, each element $[i][j]$ is 1 if there is an edge between vertex $V_i$ and vertex $V_j$, and 0 otherwise. Hence, the first matrix can be:$$\left[\begin{array}{ccccccccccccc}0& 1& 0& 1\text{1}& 0& 1& 0\text{0}& 1& 0& 1\text{1}& 0& 1& 0\end{array}\right]$$

Construct the second adjacency matrix

Now let's construct another adjacency matrix by altering labeling of vertices in the original $C_4$, such as $V_3-V_1-V_4-V_2-V_3$. The new adjacency matrix is:$$\left[\begin{array}{ccccccccccccc}0& 1& 1& 0\text{1}& 0& 0& 1\text{1}& 0& 0& 1\text{0}& 1& 1& 0\end{array}\right]$$

Construct the first adjacency list

For the first adjacency list, we'll list neighbors for each vertex.- $V_1:$ {$V_2,V_4$}- $V_2:$ {$V_1,V_3$}- $V_3:$ {$V_2,V_4$}- $V_4:$ {$V_1,V_3$}

Construct the second adjacency list

We need to represent the graph slightly differently with the new labeling for the adjacency list:- $V_1:$ {$V_3,V_4$}- $V_2:$ {$V_4,V_3$}- $V_3:$ {$V_1,V_2$}- $V_4:$ {$V_1,V_2$}

Key concepts

Adjacency Matrix

An adjacency matrix is a useful way to represent a graph using a square matrix. Each element in the matrix indicates whether pairs of vertices are adjacent or not in the graph. Let's dive deeper into how it works.

For a graph with vertices, we have an adjacency matrix with rows and columns corresponding to the vertices. When vertex $V_i$ is connected to vertex $V_j$, the element $[i][j]$ in the matrix is 1, otherwise it is 0. This simple representation makes adjacency matrices easy to understand and very efficient to use in certain computations like finding paths.

The cycle graph $C_4$ consists of 4 vertices arranged in a loop, leading to its adjacency matrix featuring diagonal zeros (no loops) and ones representing each edge:

• The first matrix, where each vertex connects in the sequence $V_1-V_2-V_3-V_4-V_1$, results in a matrix with ones across diagonizing pairs like $[1][2]$ and $[3][4]$.
• For the second matrix, changing the labeling alters the matrix's appearance but not its fundamental properties. This flexibility showcases how relabeling can produce different yet valid matrices for the same graph.

Overall, adjacency matrices offer a powerful way to visualize and work with graph data, especially for small graphs like $C_4$.

Adjacency List

An adjacency list is another way to represent a graph. Unlike the adjacency matrix, which uses space even for non-existent edges, adjacency lists are more space-efficient, listing only existing edges.

Here, each vertex is associated with a list of other vertices it shares an edge with. This representation is particularly useful for larger graphs where saving space is crucial.

For the cycle graph $C_4$, with 4 vertices ($V_1,V_2,V_3,V_4$), we see each vertex in $C_4$ connects to two other vertices in a loop:

• The first adjacency list maintains the order $V_1->V_2$ and $V_2->V_3$, rotating around the circle back to $V_1$, capturing the circular nature of a cycle.
• The second list changes the labels, but the edges still form a closed loop, illustrating the graph's cyclical form. For example, with $V_1$ also connecting to $V_3$, the cycle nature remains intact.

Each approach to listing presented edges reflects both flexibility and efficiency, with the choice of structure depending on the specific needs of computations or storage.

Cycle Graph C4

Cycle graphs like $C_4$ are a fundamental concept in graph theory. These graphs are comprised entirely of a single cycle, with each vertex connected to two others, forming a closed loop.

The simplest cycle graph, $C_4$, consists of 4 vertices: $V_1,V_2,V_3,V_4$. Each is connected sequentially in a circular pattern. This structure is characterized by:

• Equal degree: Every vertex in $C_4$ has a degree of 2, simplifying many theoretical analyses and calculations.
• Simplicity: Being both undirected and simple, the graph $C_4$ is easy to study and has no parallel edges or loops.
• Symmetry: This symmetry makes cycle graphs robust tools in theoretical studies, providing consistent templates for complex concepts such as connectivity and traversal.

Cycle graphs like $C_4$ are pivotal in understanding more intricate graphs, offering insights into connectivity and pathfinding that are critical to both theoretical studies and practical applications in computer science.