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 $1\le n\le 4$.

Recently the first and last authors defined a new graph characteristic called omega related to Euler characteristic to determine several topological and combinatorial properties of a given graph. This new characteristic is defined in terms of a given degree sequence as a graph invariant and gives a lot of information on the realizability, number of realizations, connectedness, cyclicness, number of components, chords, loops, pendant edges, faces, bridges etc. of the family of realizations. In this paper, the effect of the deletion of vertices and edges from a graph on omega invariant is studied.

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.

A realizable degree sequence can be realized in many ways as a graph. There are several tests for determining realizability of a degree sequence. Up to now, not much was known about the common properties of these realizations. Euler characteristic is a well-known characteristic of graphs and their underlying surfaces. It is used to determine several combinatorial properties of a surface and of all graphs embedded onto it. Recently, last two authors defined a number Omega which is invariant for all realizations of a given degree sequence. Omega is shown to be related to Euler characteristic and cyclomatic number. Several properties of Omega are obtained and some applications in extremal graph theory are done by authors. As already shown, the number Omega gives direct information compared with the Euler characteristic on the realizability, number of realizations, being acyclic or cyclic, number of components, chords, loops, pendant edges, faces, bridges etc. In this paper, another important topological property of graphs which is connectedness is studied by means of Omega. It is shown that all graphs with Omega(G) <= -4 are disconnected, and if Omega(G) >= -2, then the graph could be connected or disconnected. It is also shown that if the realization is a connected graph and Omega(G) = -2, then certainly the graph should be acyclic. Similarly, it is shown that if the realization is a connected graph G and Omega(G) >= 0, then certainly the graph should be cyclic. Also, the fact that when Omega(G) <= -4, the components of the disconnected graph could not all be cyclic, and that if all the components of a graph G are cyclic, then Omega(G)) >= 0 are proven.

Topological indices are very important in Mathematics and have a lot of applications in Molecular Chemistry and many other areas. About the half of them are defined in terms of vertex degrees. The degree sequence of a graph is the set of vertex degrees. It gives many information about the properties of the graph and also the real life situations that the graph corresponds. Graph operations are used to obtain larger graphs from given graphs and vice versa. Two of the most important and applicable graph operations, the join and corona product, are studied and the degree sequences of the graphs obtained by these two operations from two simple connected graphs were determined by the authors recently. Also the algebraic structure of this two graph operations were determined. In this paper, we generalize these results to the join and corona products of any number of graphs and obtain several formulae giving the degree sequence of the resulting graphs.