New concepts of vague graphs
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ORIGINAL ARTICLE
New concepts of vague graphs R. A. Borzooei1 • Hossein Rashmanlou2
Received: 31 January 2015 / Accepted: 7 December 2015 Ó Springer-Verlag Berlin Heidelberg 2015
Abstract The concept of vague graph was introduced by Ramakrishna (Int J Comput Cognit 7:51–58, 2009). Since the vague models give more precision, flexibility, and compatibility to the system as compared to the classical and fuzzy models, in this paper, the concept of energy of fuzzy graph is extended to the energy of a vague graph. It has many applications in physics, chemistry, computer science, and other branches of mathematics. We define adjacency matrix, degree matrix, laplacian matrix, spectrum, and energy of a vague graph in terms of their adjacency matrix. The spectrum of a vague graph appears in physics statistical problems, and combinatorial optimization problems in mathematics. Also, the lower and upper bounds for the energy of a vague graph are also derived. Finally, we give some applications of energy in vague graph and other sciences. Keywords graph
Vague set Vague graph Energy of vague
Mathematics Subject Classification
05C99 05C22
& Hossein Rashmanlou [email protected] R. A. Borzooei [email protected] 1
Department of Mathematics, Shahid Beheshti University, Tehran, Iran
2
Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran
1 Introduction In 1965, Zadeh [37] first proposed the theory of fuzzy sets. The most important feature of a fuzzy set is that it consists of a class of objects that satisfy a certain (or several) property. For example, for a fuzzy set A, each object x has a membership degree of A, denoted as lA ðxÞ. Gau and Buehrer [10] proposed the concept of a vague set in 1993, by replacing the value of an element in a set with a subinterval of [0, 1]. Namely, a true-membership function tv ðxÞ and a false membership function fv ðxÞ are used to describe the boundaries of the membership degree. The first definition of a fuzzy graph was proposed by Kafmann [14] in 1973, from Zadeh’s fuzzy relations [37–39]. Mordeson [16] defined fuzzy line graphs. But Rosenfeld [20] introduced another elaborated definition including fuzzy vertex and fuzzy edges and several fuzzy analogs of graph theoretic concepts such as paths, cycles, connectedness and etc. The pictorial representation of a graph consists of a set of points joined by arcs. To make use of computers to solve problems on graphs, they had to be stored in the memory of computers. This is done by using matrices. Many kinds of matrices are associated with a graph. The spectrum of one such matrix, the adjacency matrix, is called the spectrum of the graph. The properties of the spectrum of a graph is related to the properties of the graph. The area of graph theory that deals with this is called spectral graph theory. The spectrum of a graph first appeared in a paper by Collatz and Sinogowitz in 1957. At present, it is widely studied owing to its applications in physics, chemistry, computer science, and other branches
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