Abstract
Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital.
Highlights
Graph theory serves as an exceptionally beneficial tool in solving combinatorial problems in various fields, such as geometry, algebra, number theory, topology, and social systems
The construction of the paper is as follows: In Section 2 we propose the concepts of highly irregular (HI), neighborly irregular (NI), strongly irregular (SI), highly totally irregular (HTI), neighborly totally irregular (NTI), and STIVG in vague graph (VG) and study their properties
We have considered an example of a strongly irregular vague graphs (SIVGs)-G, presented in Figure 2, that is both highly irregular vague graphs (HIVGs) and neighborly irregular vague graph (NIVG)
Summary
Graph theory serves as an exceptionally beneficial tool in solving combinatorial problems in various fields, such as geometry, algebra, number theory, topology, and social systems. Fuzzy set theory is a highly powerful mathematical tool for solving approximate reasoning related problems. These notions meritoriously illustrate complicated phenomena, which are not precisely described using classical mathematics. Rashmanlou et al [29,30,31,32] introduced some properties of FGs. Sunitha et al [33,34,35] presented new concepts for fuzzy graphs. A new kind of domination set in irregular vague graphs (IVGs) was introduced and its properties were studied.
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