Math Problem Statement
Problem 1. Using the adjacency matrix, compute the number of paths of length n in the complete graph Kp from: (a) a vertex to itself; and (b) a vertex to a different vertex. Conclude a formula for the number of sequences a1a2···an ∈[p]n with ai ̸= ai+1 and an ̸= a1.
Solution
To solve the problem, let's analyze the given tasks in detail.
(a) Paths of length from a vertex to itself in :
In a complete graph , every vertex is connected to every other vertex, but there are no loops (self-loops). The adjacency matrix for has the following structure:
\begin{cases} 1 & \text{if } i \neq j, \\ 0 & \text{if } i = j. \end{cases}$$ The entry $$(A^n)_{ij}$$ in the $$n$$-th power of the adjacency matrix represents the number of paths of length $$n$$ from vertex $$i$$ to vertex $$j$$. #### Self-paths: For a vertex to return to itself after $$n$$ steps: - At each step, the vertex must move to a different vertex (since no self-loops are allowed). - There are $$p - 1$$ choices for the first step and $$p - 1$$ choices for each subsequent step. Thus, the total number of paths of length $$n$$ from a vertex to itself is: $$(p-1)^n.$$ --- ### **(b) Paths of length $$n$$ from a vertex to a different vertex:** To end at a different vertex after $$n$$ steps: - The first step offers $$p - 1$$ choices (as the vertex must move to another vertex). - For the remaining $$n-1$$ steps, there are $$p-1$$ choices at each step. Thus, the total number of paths of length $$n$$ from a vertex to a different vertex is: $$(p-1)^n.$$ This is the same as the formula for paths to the same vertex because the structure of $$K_p$$ ensures symmetry in path counting. --- ### **Concluding the Formula:** To find the number of sequences $$a_1 a_2 \cdots a_n \in [p]^n$$ such that $$a_i \neq a_{i+1}$$ for $$1 \leq i < n$$, and $$a_n \neq a_1$$, we consider: 1. The total number of paths of length $$n$$ is $$(p-1)^n$$, as derived earlier. 2. To ensure $$a_n \neq a_1$$, we subtract the cases where the sequence forms a cycle: - If $$a_1 = a_n$$, the total count is equivalent to paths returning to the starting vertex, i.e., $$(p-1)^n$$. Thus, the number of such sequences is: $$(p-1)^n - (p-1)^{n-1}.$$ --- ### **Final Result:** - **Paths of length $$n$$ from a vertex to itself:** $$(p-1)^n$$, - **Paths of length $$n$$ from a vertex to a different vertex:** $$(p-1)^n$$, - **Number of sequences $$a_1 a_2 \cdots a_n$$ with $$a_i \neq a_{i+1}$$ and $$a_n \neq a_1$$:** $$(p-1)^n - (p-1)^{n-1}.$$ --- Let me know if you want further clarifications or examples to demonstrate this! Here are some related questions for exploration: 1. How would the formula change if self-loops were allowed in $$K_p$$? 2. What happens to the formulas if $$K_p$$ is replaced by a non-complete graph? 3. Can the adjacency matrix approach be used to count paths in directed graphs? 4. How does the symmetry of $$K_p$$ simplify path counting compared to arbitrary graphs? 5. What is the relationship between adjacency matrices and eigenvalues in graph path counting? **Tip:** Understanding adjacency matrices' powers is key to solving graph-theoretic problems efficiently!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Graph Theory
Adjacency Matrix
Path Counting in Graphs
Combinatorics
Formulas
Number of paths of length n from a vertex to itself: (p-1)^n
Number of paths of length n from a vertex to a different vertex: (p-1)^n
Number of sequences a1a2···an with ai ≠ ai+1 and an ≠ a1: (p-1)^n - (p-1)^(n-1)
Theorems
Graph Adjacency Matrix Powers
Suitable Grade Level
Grades 11-12 and college level
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