Exam 10: A: Graphs

the grid graph G_m,n refers to the graph obtained by taking an m × n rectangular grid of streets (m ≤ n) with m north/south blocks and n east/west blocks. For example: -Find a formula for the number of edges of $G _ { m , n }$

Are these two digraphs isomorphic?

Find the edge chromatic number for each of the graphs -

During the construction of a home there are certain tasks that have to be completed before another one can commence, e.g., the roof has to be installed before the work on electrical wiring or plumbing can begin. How can a graph be used to model the different tasks during the construction? Should the edges be directed or undirected? Looking at the graph model, how can we find tasks that can be done at any time and how can we find tasks that do not have to be completed before other tasks can begin?

Give a recurrence relation for v_n = number of vertices of the graph $Q _ { n }$

In K₅ find the number of paths of length 3 between every pair of vertices.

either give an example or prove that there are none. -A graph with 9 vertices with edge-chromatic number equal to 2.

Construct a call graph for five friends Alice, Bob, Charlie, Diane and Evan, if there were three calls from Alice to Bob, two calls from Alice to Diane, five calls from Alice to Evan, one call from Bob to Alice, three calls from Charlie to Alice, one call from Charlie to Evan, one call from Diane to Charlie, and one call from Evan to Diane.

either give an example or prove that there are none. -A graph with 7 vertices that has a Hamilton circuit but no Euler circuit.

In K₅ find the number of paths of length 6 between every pair of vertices.

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. - $K _ { 4,5 }$