Akademik Veri Yönetim Sistemi
Mediterranean International Conference on Research In Applied Sciences, Antalya, Türkiye, 22 - 24 Nisan 2022, ss.267-268
Graph theory, where many fields come together, is an area that gives different and useful
results, and relationships have been discovered and transferred to technological developments
by the combination of methods and results that seem unconnected. The reason for the rapid
increase of interest in graph theory and its applications in recent years is that many problems
encountered in daily life can be solved with graph theory. Graphs are actually mathematical
models of real-life events. By using these models, and with the help of the theories that exist in various fields of
mathematics, the mathematical values and results obtained can be used to get an idea about
the events represented by the graphs. Affine planes are one of the most important examples of
finite geometries. A concept (or an integer) called order is defined in affine planes and with
the help of this order, the number of lines passing through a point of the plane, the number of
points on a line of the plane, also the number of total points and number of lines in the plane
are known. Recently, graphs have been obtained by the method based on the assumption that
the lines of finite affine planes are taken as a "path" and some properties of the graphs that
emerge in this way have also been examined. The most common feature among these
investigations is the degree sequences. Topological indices are constants which have a lot of
applications in Graph Theory. For the first time, they were started to be defined in 1940s, and
by the technological advances in computer technology, they are now preferred to previously
used methods. A lot of topological indices are defined for this reason. These indices are
mostly defined in terms of vertex degrees, distances or matrices corresponding to graphs. The
neighborhood(adjacency) matrix is used in many fields in graph theory and molecular
chemistry. The sum of the absolute values of the eigenvalues of the neighborhood matrix
gives the energy of a graph. Since the concept of energy has many chemical interpretations in
graph theory, it is a very useful and widely applied concept. In this study; some frequently
used topological indices in term of the vertex degrees for the affine path graphs are considered
and some general formulas for them are obtained. Finally; the vertex-adjacency matrices and
energies related to affine path graphs, that are obtained from the affine planes of order
k=2,3,4,5 are calculated