(V, E) consists of a nonempty finite set V of vertices (aka nodes) and a finite...
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(V, E) consists of a nonempty finite set V of "vertices" (aka "nodes") and a finite set E of distinct unordered pairs of A simple graph T distinct elements of V called "edges". We say that an edge {v1, v2} joins the vertices v1 and vg. If the vertices of G are labeled {1, 2,..., n}, we define the connection matrix Ar to be the n x n matrix whose i, j entry is 1 if i is joined to j with an edge, and 0 if not. for 1 < i<j<n, the (i, j) minor of the n x n matrix A is the determinant of the 2 x 2 submatrix consisting of the matrix entries with row and column given by i or j, namely aji aij Mi.5) det aji ajj 1. Prove that the sum of all the (i, j) minors of the connection matrix coincides with the negative of the number of edges in the graph I. 2. Write the connection matrix of the graph made from the vertices and edges of the octahedron, and verify the above result in this case. 3. A graph path of length k on I' is what it sounds like: a sequence (v1, ... vk) of vertices with the property that every consecutive pair (vi, vi+1) is an edge of the graph. Prove that the (i, j) entry of A4 coincides with the number of graph paths of length k from i to j. 4. Compute the number of graph paths of length k from a vertex of the cube to its antipodal vertex, for k = 1, 2,... 10. (V, E) consists of a nonempty finite set V of "vertices" (aka "nodes") and a finite set E of distinct unordered pairs of A simple graph T distinct elements of V called "edges". We say that an edge {v1, v2} joins the vertices v1 and vg. If the vertices of G are labeled {1, 2,..., n}, we define the connection matrix Ar to be the n x n matrix whose i, j entry is 1 if i is joined to j with an edge, and 0 if not. for 1 < i<j<n, the (i, j) minor of the n x n matrix A is the determinant of the 2 x 2 submatrix consisting of the matrix entries with row and column given by i or j, namely aji aij Mi.5) det aji ajj 1. Prove that the sum of all the (i, j) minors of the connection matrix coincides with the negative of the number of edges in the graph I. 2. Write the connection matrix of the graph made from the vertices and edges of the octahedron, and verify the above result in this case. 3. A graph path of length k on I' is what it sounds like: a sequence (v1, ... vk) of vertices with the property that every consecutive pair (vi, vi+1) is an edge of the graph. Prove that the (i, j) entry of A4 coincides with the number of graph paths of length k from i to j. 4. Compute the number of graph paths of length k from a vertex of the cube to its antipodal vertex, for k = 1, 2,... 10.
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Related Book For
A Concise Introduction to Logic
ISBN: 978-1305958098
13th edition
Authors: Patrick J. Hurley, Lori Watson
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