A study on cubic graphs with novel application

Abstract

Theoretical concepts of graphs are highly utilized by computer science applications. Especially in research areas of computer science such as data mining, image segmentation, clustering, image capturing and networking. The cubic graphs are more flexible and compatible than fuzzy graphs due to the fact that they have many applications in networks. In this paper, we define the direct product, strong product, and degree of a vertex in cubic graphs and investigate some of their properties. Likewise, we introduce the notion of complete cubic graphs and present some properties of self complementary cubic graphs. Finally, We present fuzzy cubic organizational model as an example of cubic digraph in decision support system.

Reinforcement Number of a Graph with respect to Half-Domination

“…Now either T e 3 (u 23 ) � 0 or T e 3 (u 12 ) � 0, and either T e 4 (v 23 ) � 0 or T e 4 (v 12 ) � 0. As A 1 � e 1 , e 2 , e 3 , e 4 ⊂ A and T A (s) ≥ (1/2) for s ∈ c, b, u 13 , u 12 , v 12 , v 13 and us, |A| ≥ 6. e upper bound 2Δ for R (1/2) (G) can be reduced further if…”

“…We can also assume that every edge e in A has one end in B. Now, we establish our claim |A| ≥ 6 by considering various possibilities for B (for each possibility, we consider one subcase; other subcases can be proved similarly): 13 , v 23 }, then |A ′ | ≥ 4 (A ′ contains at least two edges with one end at u 12 and at least two edges with one end at w 13 ). A − A ′ contains an edge e 1 with one end at w 13 and T e 1 (u 13 ) ≥ (1/2) and contains another edge e 2 with one end at u 12 and T e (w 23 ) ≥ (1/2).…”

“…Total reinforcement number of a graph has been studied in [9][10][11][12]. Recently, Muhiuddin et al have studied various related concepts on graphs (see, e.g., [13][14][15][16][17]).…”

“…13 , w 13 , v 12 , w 12 , x 2 , y 2 , z 2 } and f B (s) � 0 if s ∈ c, y 3 , z 3 , w 23 , v 23 }. e set A contains a subset A ′ , with |A ′ | � 3, T A′ (s) ≥ (1/2) for s ∈ u 12 , v13 , w 13 , and each edge in A ′ has one end at x 2 . Note that T A′ (c) ≤ (1/2), either T A′ (w 23 ) � 0 or T A′ (w 12 ) � 0, and eitherT A′ (v 12 ) � 0 or T A′ (v 23 ) � 0.…”

Preservation of the Classical Meanness Property of Some Graphs Based on Line Graph Operation

“…A classical mean labeling is for getting more accuracy of all the edge labeling by using the average of four different types of means of the vertex labeling of the given graph. Recently, Muhiuddin et al studied various related concepts on graphs (see [18][19][20][21][22]).…”