In this note, we find all the solutions of the Diophantine equation x 2 + 2a · 3b · 11c = y n, in nonnegative integers a, b, c, x, y, n ≥ 3 with x and y coprime.

Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many applications in different areas from Biology, Architecture, Anatomy to Finance. In this paper, we define Fibonacci graphs as graphs having degree sequence consisting of n consecutive Fibonacci numbers and use the invariant $\Omega$ to obtain some more information on these graphs. We give the necessary and sufficient conditions for the realizability of a set D of n successive Fibonacci numbers for every n and also list all possible realizations called Fibonacci graphs for $1\le n\le 4$.

A derived graph is a graph obtained from a given graph according to some predetermined rules. Two of the most frequently used derived graphs are the line graph and the total graph. Calculating some properties of a derived graph helps to calculate the same properties of the original graph. For this reason, the relations between a graph and its derived graphs are always welcomed. A recently introduced graph index which also acts as a graph invariant called omega is used to obtain such relations for line and total graphs. As an illustrative exercise, omega values and the number of faces of the line and total graphs of some frequently used graph classes are calculated.

Average degree of a graph is defined to be a graph invariant equal to the arithmetic mean of all vertex degrees and has many applications, especially in determining the irregularity degrees of networks and social sciences. In this study, some properties of average degree have been studied. Effect of vertex deletion on this degree has been determined and a new proof of the handshaking lemma has been given. Using a recently defined graph index called $omega$ index, average degree of trees, unicyclic, bicyclic, and tricyclic graphs have been given, and these have been generalized to $k$ -cyclic graphs. Also, the effect of edge deletion has been calculated. The average degree of some derived graphs and some graph operations have been determined.

Special numbers have very important mathematical properties alongside their numerous applications in many fields of science. Probably the most important of those is the Fibonacci numbers. In this paper, we use a generalization of Fibonacci numbers called tribonacci numbers having very limited properties and relations compared to Fibonacci numbers. There is almost no result on the connections between these numbers and graphs. A graph having a degree sequence consisting oftsuccessive tribonacci numbers is called a tribonacci graph of ordert. Recently, a new graph parameter named as omega invariant has been introduced and shown to be very informative in obtaining combinatorial and topological properties of graphs. It is useful for graphs having the same degree sequence and gives some common properties of the realizations of this degree sequence together with some properties especially connectedness and cyclicness of all realizations. In this work, we determined all the tribonacci graphs of any order by means of some combinatorial results. Those results should be useful in networks with large degree sequences and cryptographic applications with special numbers.