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GRAPH-THEORETIC MODELS I

In Chapter 20, we used the Q-matrix to extract some interesting properties of both Fibonacci and Lucas numbers. We will now employ it to construct graph-theoretic models for them, and then explore some of the well-known identities. These models enable us to study Fibonacci and Lucas numbers in a different perspective.

To begin with, recall that

equation

where c021-math-001.

We are now ready for the models.

21.1 A GRAPH-THEORETIC MODEL FOR FIBONACCI NUMBERS

The Q-matrix can geometrically be translated into a connected graph G with two vertices c021-math-002 and c021-math-003; see Figure 21.1. In fact, Q is the adjacency matrix c021-math-004 of the graph, where c021-math-005 denotes the number of edges from c021-math-006 to , and .

Figure 21.1

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