On the novel existence results of solutions for a class of fractional boundary value problems on the cyclohexane graph

A branch of mathematical science known as chemical graph theory investigates the implications of connectedness in chemical networks. A few researchers have looked at the solutions of fractional differential equations using the concept of star graphs. Their decision to use star graphs was based on the assumption that their method requires a common point linked to other nodes but not to each other. Our goal is to broaden the scope of the method by defining the idea of a cyclohexane graph, which is a cycloalkane with the molecular formula C6H12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{6}H_{12}$\end{document} and CAS number 110-82-7. It consists of a ring of six carbon atoms, each bonded with two hydrogen atoms above and below the plane with multiple junction nodes. This article examines the existence of fractional boundary value problem’ solutions on such graphs in the sense of the Caputo fractional derivative by using the well-known fixed point theorems. In addition, an example is given to support our key findings.

Chemical graph theory is one of the fields of mathematics that examines the consequences of a chemical network's connectivity. A chemical graph may represent any actual or abstract chemical system (i.e., molecular transformations in a chemical reaction). Chemical graph theory, in other words, is focused on every element of graph theory's application to chemistry. Furthermore, in chemical graph theory, the word "chemical" is used to highlight that, unlike graph theory, one may depend on scientific observation of many ideas and theorems rather than rigorous mathematical proofs, which is a crucial difference. Lumer's research [20] was the first to look at differential equation theory applied to graphs. By altering specified local operators, he investigated extended equations of evolution on ramification spaces. In 1989, using a geometric network, Zavgorodnij [21] studied linear differential equations, with the existence of suggested boundary value problem solutions arranged at the interior vertices of the network. Gordeziani et al., on the other hand, used the double-sweep technique to find the numerical solutions for ordinary differential equations on graphs, which they found to be more efficient (see [22]).
However, just a few studies on fractional boundary value problems with graphs have shown the existence of solutions using specific fixed point techniques (see [23,24]). The concept of a star graph was used by the authors of these works, namely are sets of vertices and edges, respectively, such thatp 0 is the junction node andr = − − → p p 0 is the edge connecting nodesp top 0 having length˜ = | − − → p p 0 | for = 1, 2 (see Fig. 1). After establishing a local coordinate system with the origin at the verticesr 1 andr 2 , the coordinate s ∈ (0,˜ ) is investigated on each edger = − − → p p 0 (as seen in Fig. 1). Graef et al. [23] suggested the subsequent system of nonlinear fractional differential equations defined on each edger = − − → p p 0 and applied the notable fixed point theorems to prove the existence of solutions of the following problem: and also L : [0,˜ ] × R → R are continuous functions. Also, RL D j and RL D k represent Riemann-Liouville fractional derivatives of orders j and k, respectively. In [24], Mehandiratta et al. broaden the above work to (u + 1) vertices with where the length of eachr joining verticesp top 0 ( = 1, 2, . . . , u) is˜ = | − − → p p 0 |. They investigated the solutions of the following problem: ⎧ ⎨ ⎩ D j w (s) = H (s, w (s), D k w (s)) (s ∈ (0,˜ ), = 1, 2, . . . , u), Recently, Mophou et al. [25] investigated the solution of the following fractional Sturm-Liouville boundary value problems on a star graph: where D j a + and D j b -1 , = 1, 2, . . . , n stand, respectively, for the left Riemann-Liouville and the right Caputo fractional derivative of order j ∈ (0, 1); I j a + is the Riemann-Liouville fractional integral of order j. The real functions β and q are defined on [a, b ] ( = 1, 2, . . . , n). The function H belongs to L 2 (a, b ), = 1, 2, . . . , n, and the controls v , = 1, 2, . . . , n are real variables.
For the recent research in this area, we refer to [26][27][28][29] and the references therein.
In this work, we utilized the concept of the cyclohexane graph (see Fig. 2) to extend the idea of the above fractional boundary value problems to a new problem that is more generic than star graphs.
The method used in [23] and [24] for identifying the origin at boundary nodes other than the junction nodep 0 would be insufficient since a cyclohexane graph has several junction points. As a result, we use another procedure in which we label the vertices of the preceding graph with 0 or 1 having edge length˜ = 1 (see Fig. 3).
Here, we examine the existence of solutions to the following problem: where η κ (κ = 1, 2, 3) are real constants with η κ = 0, D j , D j-1 , and D k are the Caputo fractional derivative of orders j ∈ (1, 2], j -1 ∈ (0, 1], and k ∈ (0, 1), respectively. Also, where is the number of edges of the graph representation of cyclohexane compound with |r k | = 1. Also, D 2m is the sequential fractional derivative discussed in [2] ⎧ ⎨ ⎩ D k w = D k w, Our goal is to use the relevant fixed point theorems to prove the existence of solutions to the proposed problem (1.6). Finally, an example is given to highlight the significance of our results in the particular literature.

Preliminaries
In the next sections, the subsequent outcomes will be required.
where [j] represents the integer part of j.

Lemma 2.3 Suppose that real-valued functions
if and only if w is a solution for the following fractional integral equation: Proof Suppose that w is a solution of (2.1), where = 1, 2, . . . , 18. Then there exist con- Using the boundary conditions for (2.1), we have Substituting the values of z ( ) 0 and z ( ) 1 into (2.3), we obtain the solution (2.2). With regard to the converse statement, it is self-evident that w is a solution for (2.1) when it is a solution for an integral equation (2.3).
The fixed point theorems of Schaefer and Krasnoselskii are now presented.

Theorem 2.4 ([30]) If T is a completely continuous, that is, T is continuous and totally
bounded, self-operator on a Banach space C, then either {a ∈ C : a = bT a for some b ∈ (0, 1)} is unbounded or T has a fixed point.
for all k, k ∈ O. Then T 1 + T 2 has a fixed point.

Main results
We define the Banach space C = {w : w , D k w , w ∈ C[0, 1]} with the norm 1] w (s) for = 1, 2, . . . , 18. It is obvious that the product space C = C 1 × C 2 × · · · × C 18 is a Banach space, where the norm is defined by In order to apply Lemma 2.3, we introduce the operator T : C → C by where for all s ∈ [0, 1] and w ∈ C .
To simplify calculations, we shall use the following notation:

5)
(3.10) Proof It is obvious from the implication of (3.2) that the fixed points of T defined by (3.1) exist if and only if (1.6) has a solution. To prove this, we first show that T is completely continuous.
We shall now examine the solution of the fractional boundary value problem (1.6) by applying various conditions.
To show the significance of our results, we present the following example.
of solutions to the suggested problem. Our method is easy to implement and may be used on a wide range of graphs, including chordal bipartite graphs, which have many applications in computer networking and biology.