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Much work has been done related to these graph operations.
Lan and Zhou [12] defined four new graph operations based on R-graphs and determined the adjacency (respectively, Laplacian and singles Laplacian) spectra of these graph operations.
Khalifeh et al. [15] gave the exact formulas for the Zagreb indices of several graph operations.
Structures can be newly created, selected, adapted or merged by using the graph operations.
There are two main kinds of graph operations: unary and binary.
In this section, some exact formulae for the Harary index of some graph operations are presented.
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Before this, we introduce a new graph operation as follows.
Godsil and McKay [13] introduced a new graph operation (rooted product) and then found its spectrum.
It was not until much later that formal languages were used to represent biomedical ontologies and recast the graph operation in terms of deductive inference over formal theories [ 19 22].
All mathematical and graphing operations were performed using MATLAB-7.4.0(R2007a) and Microsoft Office Excel 2003 programs.
The population is first initialized with consistent labelings (graphs) and evolved exploiting the graph-union and graph-intersection operations between individuals.
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