Topological Indices of Some New Graph Operations and Their Possible Applications

Abstract

A chemical graph theory is a fascinating branch of graph theory which has many applications related to chemistry. A topological index is a real number related to a graph, as its considered a structural invariant. It’s found that there is a strong correlation between the properties of chemical compounds and their topological indices. In this paper, we introduce some new graph operations for the first Zagreb index, second Zagreb index and forgotten index "F-index". Furthermore, it was found some possible applications on some new graph operations such as roperties of molecular graphs that resulted by alkanes or cyclic alkanes.

Some K-Banhatti Polynomials of First Dominating David Derived Networks

“…2) 𝐵 1 (𝐺, 𝑥, 𝑦) = 4𝑚 𝑥 4 𝑦 4 + (4𝑚 − 4) 𝑥 5 The K Banhatti polynomial sum connectivity 𝐷 1 (𝑚) is (as shown in Fig. 5) If 𝐺 is the DDD network, then 𝐵 2 𝑎 (𝐺, 𝑥, 𝑦) = 4𝑚 𝑥 (4) 𝑎 𝑦 (4) 𝑎 + (4𝑚 − 4) 𝑥 (6) 𝑎 𝑦 (9) 𝑎 + (28𝑚 − 16)𝑥 (8) 𝑎 𝑦 (16) 𝑎 + (9𝑚 2 − 13𝑚 + 24) 𝑥 (12) 𝑎 𝑦 (12) 𝑎 + (36𝑚 2 − 56𝑚 + 24)𝑥 (15) 𝑎 𝑦 (20) 𝑎 + (36𝑚 2 − 56𝑚 + 20)𝑥 (24) 𝑎 𝑦 (24) 𝑎 Proof: In aforesaid Table 1. +(36𝑚 2 − 56𝑚 + 20)𝑥 (4 * 6) 𝑎 𝑦 (4 * 6) 𝑎 = 4𝑚 𝑥 (4) 𝑎 𝑦 (4) 𝑎 + (4𝑚 − 4) 𝑥 (6) 𝑎 𝑦 (9) 𝑎 + (28𝑚 − 16)𝑥 (8) 𝑎 𝑦 (16) 𝑎 + (9𝑚 2 − 13𝑚 + 24) 𝑥 (12) 𝑎 𝑦 (12) 𝑎 + (36𝑚 2 − 56𝑚 + 24)𝑥 (15) 𝑎 𝑦 (20) 𝑎 + (36𝑚 2 − 56𝑚 + 20)𝑥 (24) 𝑎 𝑦 (24) 𝑎 The following are the results of Theorem 2.…”

“…The second K Banhatti polynomial 𝐷 1 (𝑚) is (as shown in Fig. 6) 𝐵 2 (𝐺, 𝑥, 𝑦) = 4𝑚 𝑥 4 𝑦 4 + (4𝑚 − 4) 𝑥 6 The second hyper K Banhatti polynomial 𝐷 1 (𝑚) is (as shown in Fig. 7) Proof: Using Table 1.…”

“…Theorem 1: = ∑ 𝐸 𝑢𝑒 𝑥 [𝑑 𝐺 (𝑢)+𝑑 𝐺 (𝑒)] 𝑎 𝑦 [𝑑 𝐺 (𝑣)+𝑑 𝐺 (𝑒)] 𝑎 𝑢𝑒 𝑑(𝑢)≤𝑑(𝑣) = 4𝑚 𝑥 (2+2) 𝑎 𝑦 (2+2) 𝑎 + (4𝑚 − 4) 𝑥 (2+3) 𝑎 𝑦 (3+3) 𝑎 + (28𝑚 − 16)𝑥 (2+4) 𝑎 𝑦 (4+4) 𝑎 +(9𝑚 2 − 13𝑚 + 24) 𝑥 (3+4) 𝑎 𝑦 (3+4) 𝑎 + (36𝑚 2 − 56𝑚 + 24)𝑥 (3+5) 𝑎 𝑦 (4+5) 𝑎 +(36𝑚 2 − 56𝑚 + 20)𝑥 (4+6) 𝑎 𝑦 (4+6) 𝑎 = 4𝑚 𝑥 (4) 𝑎 𝑦 (4) 𝑎 + (4𝑚 − 4) 𝑥 (5) 𝑎 𝑦 (6) 𝑎 + (28𝑚 − 16)𝑥 (6) 𝑎 𝑦 (8) 𝑎 +(9𝑚 2 − 13𝑚 + 24)𝑥 (7) 𝑎 𝑦 (7) 𝑎 + (36𝑚 2 − 56𝑚 + 24)𝑥 (8) 𝑎 𝑦 (9) 𝑎 +(36𝑚 2 − 56𝑚 + 20)𝑥 (10) 𝑎 𝑦 (10) 𝑎 The following are the results of Theorem 1.…”

“…A topological index is a number that describes a topological graph. This index was studied by both Mathematicians and chemists 6 . Discusses how to construct 7 an 𝑚 dimensional David derived and DDD.…”

“…Figure 5. The First sum connectivity K Banhatti polynomial of Domination David Derived Network Theorem 2:If 𝐺 is the DDD network, then 𝐵 2 𝑎 (𝐺, 𝑥, 𝑦) = 4𝑚 𝑥(4) 𝑎 𝑦 (4) 𝑎 + (4𝑚 − 4) 𝑥(6) 𝑎 𝑦(9) 𝑎 + (28𝑚 − 16)𝑥(8) 𝑎 𝑦 (16) 𝑎 + (9𝑚 2 − 13𝑚 + 24) 𝑥(12) 𝑎 𝑦(12) 𝑎 + (36𝑚 2 − 56𝑚 + 24)𝑥 (15) 𝑎 𝑦 (20) 𝑎 + (36𝑚 2 − 56𝑚 + 20)𝑥 (24) 𝑎 𝑦 (24) 𝑎 Proof: In aforesaid Table1. 𝐵 2 𝑎 (𝐺, 𝑥, 𝑦)…”