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Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with  uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2,   T = A · D = A3,   Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V   Tu,u

Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with  uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2,   T = A · D = A3,   Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V   Tu,u

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
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Discrete mathematics.

Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with  uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2,   T = A · D = A3,   Q = A · T = A4. Consider the values on the diagonals.

Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V   Tu,u

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