1. Suppose G is 26-regular on 105 vertices. Assume that for any two adjacent vertices, they share 13 neighbors, and for

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1 Suppose G Is 26 Regular On 105 Vertices Assume That For Any Two Adjacent Vertices They Share 13 Neighbors And For 1 (139.87 KiB) Viewed 35 times

1. Suppose G is 26-regular on 105 vertices. Assume that for any two adjacent vertices, they share 13 neighbors, and for any two non-adjacent vertices, they share 4 common neighbors. (a) If A is the adjacency matrix for the graph, determine the entries of AP in 3 cases for (i, j): When i = j, when i is adjacent to j, and when i is NOT adjacent to j. (b) Express A’ in terms of A, I, and all ones matrix J. (C) We know k = 26 is an eigenvalue with multiplicity 1. If v is any other eigenvector (not associated with k = 26), determine all other distinct eigenvalues by applying v to the equation in the previous step. (d) Set up two equations and solve them to determine the multiplicities of the re- maining eigenvalues.