Explore BrainMass

Graph theory: Find matrices representing the linear transformations ∂ and δ.

This content was STOLEN from BrainMass.com - View the original, and get the solution, here!

Please see attached file for full problem description.

Let C0 ={SUM (i = 1 to p) εivi│εi is an element of F2, vi is an element of V(G)}
be the vector space of 0-chains
Let C1 ={SUM (i = 1 to q) εiei│εi is an element of F2, ei is an element of E(G)}
be the vector space of 1-chains

Recall the linear transformations
boundry ∂ : C1 → C0 defined by ∂(uv) = u + v and
coboundary δ: C0 → C1 defined by δ(u) = SUM ei, where ei is adjacent to v.

Let Z(G) = { x an element of C1│∂(x) = 0} be the cycle space of G
Let B(G) ={ x an element of C1│there exists y an element of C0, x = ∂(y)} be the coboundary space of G.

a. Find matrices representing the linear transformations ∂ and δ.
b. Define an inner product on C1 by < x,y > = SUM &#949;i&#951;i, where x = SUM &#949;iei and y = SUM &#951;iei.
Prove that x is an element of Z(G) iff < x,y > = 0 for all y element of B(G)
c. Show that the dimensions of B(G) is p - k(G).
d. Characterize the class of graphs for which B(G) = C1(G)

© BrainMass Inc. brainmass.com September 19, 2018, 5:01 pm ad1c9bdddf - https://brainmass.com/math/linear-transformation/graph-theory-representing-linear-transformations-148433


Solution Preview

The explanations are in the attached pdf file.

As required by Brainmass, I past the original text in plain TEXT below.
You do not have to read it since all you need is in the pdf file.

Here is the plain TEXT source

parindent = 0pt
parskip = 12pt


centerline{bf Boundary and Coboundary}

bf (a) rm

A matrix representing the boundary operator $p$ for a gragh with $p$ vertices and $q$ edges, for $C_0$ and $C_1$ defined on field $F_2={ 0 ~ 1}$, is a $ptimes q$ matrix $M_p$ of $p$ rows and $q$ columns contaning 0s or 1s.
Each column represents an edge. If this edge connects vertices $i$ and $j$, ...

Solution Summary

Matrices and linear transformations are investigated.The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.