A simple graph I (V, E) consists of a nonempty finite set V of vertices (aka...

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A simple graph I (V, E) consists of a nonempty finite set V of "vertices" (aka "nodes") and a finite set E of distinct unordered pairs of distinct elements of V called "edges". We say that an edge {v1, v2} joins the vertices v₁ and v2. = 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 į 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 aij 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 A 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. M(i,j) = det + aji A simple graph I (V, E) consists of a nonempty finite set V of "vertices" (aka "nodes") and a finite set E of distinct unordered pairs of distinct elements of V called "edges". We say that an edge {v1, v2} joins the vertices v₁ and v2. = 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 į 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 aij 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 A 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. M(i,j) = det + aji

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Solution kle have connection Matrix A such that matrix whouse entryof A is ... View the full answer