Counting Polynomials in Chemistry: Past, Present, and Perspectives

Abstract

Counting polynomials find their way into chemical graph theory through quantum chemistry in two ways: as approximate solutions to the Schrödinger equation or by storing information in a mathematical form and trying to find a pattern in the roots of these expressions. Coefficients count how many times a property occurs, and exponents express the extent of the property. They help understand the origin of regularities in the chemistry of specific classes of compounds. Our objective is to accelerate the research of newcomers into chemical graph theory. One problem in understanding these concepts is in the different approaches and notations of each research study; some researchers provide online tools for computing these mathematical concepts, but these need to be maintained for functionality. We take advantage of similar mathematical aspects of 14 such polynomials that merge theoretical chemistry and pure mathematics; give examples, differences, and similarities; and relate them to recent research.

The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle

“…Then, D i forms a D i -decomposition of Then, D i forms a D i -decomposition of 4 K * 15 for i ∈ [2,18].…”

“…Graphs, digraphs, and multigraphs are fundamental mathematical structures of great significance in various fields, including computer science, logistics, chemistry, and biology [1][2][3][4]. In particular, digraphs have a wide range of real-world applications such as social networks, communication networks, electrical circuit design, network flow analysis, and biological networks (see [5]).…”

“…Let the vertex set of K * 9 be Z 9 and letD 2 = D 2 [0,1,2,3,4,5,6,7], D 2[1,3, 0,5,2,4,7,8], D 2 [1, 4, 0, 3, 6, 2, 8, 5], D 2 [3, 8, 7, 6, 0, 2, 1, 5], D 2 [4, 7, 3, 8, 2, 5, 0, 6], D 2 [5, 6, 1, 8, 4, 3, 2, 7], D 2 [6, 1, 3, 7, 2, 0, 4, 8], D 2 [7, 5, 4, 2, 6, 8, 0, 1],D 4 = i∈Z 9 D 4 [0, 1, 3, 2, 7, 4, 8, 6] + i , D 5 = i∈Z 9 D 5 [0, 1, 3, 8, 7, 4, 2, 5] + i , D 6 = i∈Z 9 D 6 [0, 1, 3, 8, 5, 4, 2, 6] + i , D 7 = D 7 [0,1, 2, 3, 4, 5, 6, 7], D 7 [0, 2, 1, 3, 5, 4, 6, 8], D 7 [1, 8, 3, 7, 4, 2, 6, 0], D 7 [2, 7, 5, 6, 3, 0, 4, 8], D 7 [3, 0, 4, 1, 5, 2, 7, 8], D 7 [3, 2, 6, 0, 8, 7, 5, 1], D 7 [6, 3, 5, 2, 8, 4, 1, 7], D 7 [6, 8, 5, 0, 2, 4, 7, 1], D 7 [7, 3, 4, 6, 1, 8, 5, 0] , D 8 = i∈Z 9 D 8 [0, 1, 2, 4, 6, 3, 8, 5] + i , D 9 = i∈Z 9 D 9 [0, 1, 2, 5, 3, 8, 4, 6] + i , D 10 = D 10 [0, 1, 2, 3, 4, 5, 6, 7], D 10 [1, 0, 8, 5, 7, 4, 2, 3], D 10 [2, 0, 4, 3, 1, 8, 7, 6], D 10 [4, 8, 2, 5, 0, 7, 1, 6], D 10 [6, 0, 5, 3, 7, 2, 4, 8], D 10 [6, 3, 0, 2, 8, 1, 5, 4], D 10 [7, 1, 4, 0, 6, 3, 8, 5], D 10 [7, 4, 1, 5, 2, 6, 8, 3], D 10 [8, 0,…”

“…Let the vertex set of 4 K * 11 be Z 11 and let D 2 = i∈Z 11 3(D 2 [0, 1, 2, 4, 7, 3, 8, 5] + i), D 2 [0, 1, 3, 7, 4, 8, 6, 2] + i, D 2 [0, 7, 1, 2, 6, 4, 10, 8] + i , D 3 = i∈Z 11 3(D 3 [0, 1, 6, 2, 3, 5, 10, 8] + i), D 3 [0, 3, 2, 1, 5, 9, 6, 4] + i, D 3 [0, 5, 2, 4, 1, 6, 3, 7] + i , D 4 = i∈Z 11 3(D 4 [0, 1, 2, 7, 3, 5, 8, 6] + i), D 4 [0, 3, 2, 1, 5, 9, 6, 4] + i, D 4 [0, 5, 2, 10, 1, 6, 3, 7] + i , D 5 = i∈Z 11 3(D 5 [0, 1, 2, 4, 9, 3, 10, 8] + i), D 5 [0, 3, 2, 6, 5, 1, 9, 7] + i, D 5 [0, 5, 2, 4, 1, 6, 3, 7] + i , D 6 = i∈Z 11 3(D 6 [0, 1, 2, 4, 9, 3, 10, 8] + i), D 6 [0, 3, 1, 5, 9, 8, 7, 4] + i, D 6 [0, 5, 2, 4, 1, 9, 3, 7] + i , D 7 = i∈Z 11 3(D 7 [0, 1, 7, 6, 2, 4, 10, 8] + i), D 7 [0, 1, 3, 2, 6, 10, 7, 4] + i), D 7 [0, 3, 6, 2, 4, 7, 1, 5] + i , D 8 = i∈Z 11 3(D 8 [0, 1, 3, 2, 9, 4, 7, 5] + i), D 8 [0, 1, 5, 4, 8, 10, 6, 3] + i, D 8 [0, 3, 5, 1, 4, 7, 2, 6] + i , D 9 = i∈Z 11 3(D 9 [0,1, 3, 2, 9, 4, 10, 8] + i), D 9 [0, 2, 1, 5, 6, 3, 10, 7] + i, D 9 [0, 3, 5, 1, 4, 10, 2, 6] + i , D 10 = i∈Z 11 3(D 10 [0, 1, 3, 7, 6, 8, 2, 5] + i), D 10 [0, 1, 3, 6, 2, 9, 10, 7] + i, D 10 [0, 3, 1, 6, 2, 5, 10, 7] + i , D 11 = i∈Z 11 3(D 11 [0, 1, 3, 7, 2, 5, 4, 6] + i), D 11 [0, 1, 4, 6, 2, 3, 10, 7] + i, D 11 [0, 3, 1, 8, 2, 5, 10, 7] + i , D 12 = i∈Z 11 3(D 12 [0, 1, 3, 4, 9, 5, 8, 6] + i), D 12 [0, 1, 4, 8, 10, 6, 7, 3] + i, D 12 [0, 3, 1, 4, 9, 2, 10, 6] + i , D 13 = i∈Z 11 3(D 13 [0, 1, 3, 4, 7, 5, 10, 6]+ i), D 13 [0, 1, 4, 8, 5, 7, 3, 10] + i, D 13 [0, 3, 1, 4, 8, 2, 9, 6] + i , D 14 = i∈Z 11 3(D 14 [0, 1, 2, 4, 6, 3, 10, 5] + i), D 14 [0, 1, 2, 9, 6, 8, 4, 7] + i, D 14 [0, 3, 5, 1, 9, 4, 10, 7] + i , D 15 = i∈Z 11 3(D 15 [0, 1, 7, 3, 2, 4, 10, 8] + i), D 15 [0, 2, 1, 5, 6, 3, 7, 4] + i, D 15 [0, 3, 6, 1, 8, 2, 4, 7] + i , D 16 = i∈Z 11 3(D 16 [0, 1, 7, 2, 4, 3, 10, 8] + i), D 16 [0, 2, 1, 5, 9, 10, 6, 3] + i, D 16 [0, 3, 6, 8, 2, 9, 1, 5] + i , D 17 = i∈Z 11 3(D 17 [0, 1, 3, 2, 5, 9, 4, 6] + i), D 17 [0, 1, 3, 6, 2, 9, 10, 7] + i, D 17 [0, 3, 1, 6, 9, 5, 2, 7] + i , D 18 = i∈Z 11 3(D 18 [0, 1, 3, 4, 2, 5, 10, 6] + i), D 18 [0, 1, 4, 8, 10, 6, 2, 3] + i, D 18 [0, 3, 1, 4, 9, 6, 2, 7] + i .Then, D i forms a D i -decomposition of 4 K * 11 for i ∈[2,18]. Let the vertex set of 4 K * 14 be Z 13 ∪ {∞} and letD 2 = i∈Z 13 4(D 2 [0, 1, 2, 4, 7, 11, 3, ∞] + i), 2(D 2 [0, 6, 1, 7, 2, 9, 5, 3] + i), D 2 [0,3, 1, 7, 4, 10, 6, 2] + i , D 8 = i∈Z 13 4(D 8 [0, 1, 2, 10, 12, 8, 11, ∞] + i), 2(D 8 [0, 2, 8, 1, 7, 11, 6, 3] + i), D 8 [0, 5, 1, 7, 2, 8, 6, 4] + i , D 9 = i∈Z 13 4(D 9 [0, 1, 2, 11, 3, 6, 4, ∞] + i), 2(D 9 [0, 2, 8, 1, 9, 6, 10, 3] + i), D 9 [0, 5, 1, 7, 3, 11, 4, 2] + i , D 10 = i∈Z 13 4(D 10 [0, 1, 9, 5, 3, 4, 7, ∞] + i), 2(D 10 [0, 2, 7, 1, 6, 3, 10, 4] + i), D 10 [0, 2, 5, 9, 3, 1, 7, 4] + i , D 11 = i∈Z 13 4(D 11 [0, 1, 9, 6, 8, 12, 11, ∞] + i), 2(D 11 [0, 2, 6, 12, 7, 1, 8, 3] + i), D 11 [0, 2, 5, 9, 6, 4, 11, 7] + i , D 12 = i∈Z 13 4(D 12 [0, 1, 9, 10, 7, ∞, 2, 11] + i), 2(D 12 [0, 2, 6, 1, 7, 4, 11, 5] + i), D 12 [0, 2, 5, 1, 8, 6, 3, 7] + i , D 13 = i∈Z 13 4(D 13 [0, 1, 9, 10, 12, 8, 5, ∞] + i), 2(D 13 [0, 2, 7, 1, 10, 4, 9, 3] + i), D 13 [0, 2, 5, 1, 12, 3, 10, 7] + i , D 14 = i∈Z 13 2(D 14 [0, 1, 2, 10, 12, 8, 1, ∞] + i), 2(D 14 [0, 1, 3, 2, 7, 4, ∞, 9] + i), 2(D 14 [0, 2, 8, 1, 11, 3, 12, 5] + i), D 14 [0, 2, 8, 11, 7, 1, 12, 3] + i , D 15 = i∈Z 13 4(D 15 [0, 1, 9, 10, 6, 8, 11, ∞] + i), 2(D 15 [0, 2, 5, 1, 8, 3, 9, 6] + i), D 15 [0, 5, 11, 6, 12, 8, 4, 2] + i , Let the vertex set of 4 K * 15 be Z 15 and let D 2 = i∈Z 15 4(D 2 [0, 6, 1, 5, 4, 9, 2, 8] + i), 2(D 2 [0, 2, 3, 5, 1, 13, 9, 12] + i), D 2 [0, 2, 3, 1, 13, 10, 12, 14] + i , D 3 = i∈Z 15 4(D 3 [0, 1, 6, 10, 2, 8, 13, 7] + i), 2(D 3 [0, 3, 2, 4, 7, 5, 1, 12] + i), D 3 [0, 3, 2, 4, 1, 14, 12, 13] + i , D 4 = i∈Z 15 4(D 4 [0, 1, 5, 10, 2, 7, 13, 6] + i), 2(D 4 [0, 3, 5, 4, 2, 13, 1, 12] + i), D 4 [0, 3, 5, 4, 1, 14, 12, 13] + i , D 5 = i∈Z 15 4(D 5 [0, 1, 5, 12, 2, 7, 13, 6] + i), 2(D 5 [0, 3, 1, ...…”

“…Lemma 3.For each i ∈[2,18] and each y ∈ {8, 10, 12, 14}, there exists D i -decomposition of K * 8,y .Proof.…”