علي عبد الحسين ناصر شكر

To gain better understanding of molecules, social nods, urban planning and other networks, we need to represent them as graphs which is why graph theory is a very active area in mathematics. In addition, it is well known that the Fibonacci sequence appears in many different areas. Recently, the Fibonacci and Lucas graphs were introduced and some of their algebraic properties were established. In this work, we consider generalized Lucas graphs and establish some algebraic properties. Studying the generalized Lucas graphs yields a wider class of graphs since the Fibonacci and Lucas graphs are special cases of the generalized Lucas graphs.

Its well known that there are a limited number of method which associate group theory to graph theory. In this paper, we propose a new method and obtain a novel kind of graphs. Some important application like graph energy was established to the proposed graph.

For continuous functions f with zero mean on the circle we consider the Birkhoff sums f(n, x, h) generated by the rotations by 2πh, where h is an irrational number. The main result asserts that the growth rate of the sequence maxx f(n, x, h) as n → ∞ depends only on the uniform convergence to zero of the Birkhoff means 1 n f(n, x, h). Namely, we show that for any sequence σk → 0 and any irrational h there exists a function f such that the sequence maxx f(n, x, h) increases faster than nσn. We also show that for any function f that is not a trigonometric polynomial there exist irrational h for which some subsequence maxx f(nk, x, h) increases faster than the corresponding subsequence nkσnk . We present applications to weighted shift operators generated by irrational rotations and to their resolvents. Namely, we show that the resolvent of such an operator can increase arbitrarily fast in approaching the spectrum.

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