Consider the Adjacency Matrix for a Graph That Is Shown $v _ { 1 }$

question 8

Essay

Consider the adjacency matrix for a graph that is shown below. Answer the following questions by examining the matrix and its powers only, not by drawing the graph. Show your work in a way that makes your reasoning clear.
(a) How many walks of length 2 are there from $v _ { 1 }$ to $v _ { 2 }$ ?
(b) How many walks of length 2 are there from $v _ { 1 }$ to $v _ { 3 }$ ?
(c) How many walks of length 2 are there from $v _ { 2 }$ to $v _ { 2 }$ ?

---

Definitions:

Idle Capacity

Unused or unutilized production capacity within a business, often leading to inefficiency and increased costs.

Constrained Resource

A limited resource that restricts a company's ability to meet demand, such as limited machine time or labor hours.

Automated Factory

An automated factory utilizes advanced machinery and computer systems to perform tasks with minimal human intervention, increasing efficiency and production.

Porcelain Dip Bowl

A specific type of serving bowl made from porcelain, typically used for dipping sauces or small appetizers.