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We give a total graph interpretation of the numbers of the Fibonacci type. This graph interpretation relates to an edge colouring by monochromatic paths in graphs. We will show that it works for almost all numbers of the Fibonacci type. Moreover, we give the lower bound and the upper bound for the number of all ( 1 , 2 1 )-edge colourings in trees.

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In this paper we shall show applications of the Fibonacci numbers in edgecoloured trees. In particular we determine the successive extremal graphs in the class of trees with respect to the number of (A, 2B)-edge colourings. We show connections between these numbers and Fibonacci numbers as well as the telephone numbers.

In this paper, we give the necessary and sufficient conditions for the existence of strong (1, 1, 2)-kernels in the corona of graphs. Moreover, we consider lower and upper strong (1, 1, 2)-kernel numbers and we prove that the difference between these parameters can be arbitrarily large.

In this paper, we introduce a new kind of generalized Fibonacci polynomials in the distance sense. We give a direct formula, a generating function and matrix generators for these polynomials. Moreover, we present a graph interpretation of these polynomials, their connections with Pascal’s triangle and we prove some identities for them.

In this paper we introduce a new generalization of telephone numbers. We give the generating function, direct formulae, and matrix generators for these numbers. Moreover, we present their interpretations and we prove some properties of these numbers connected with congruences.

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