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 $C_{6}H_{12}$ C 6 H 12 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.

Relaxed Skolam Mean Labeling of 5 – Star Graphs with Partition 3, 2

In this article, our main topic is about the existence of relaxed skolem mean labeling for a 5 – star graph G = K1,α ∪ K1,α ∪ K1,α ∪ K1, β ∪ K1, β 1 2 3 1 2 with partition 3, 2 with a certain condition. By using the trial and error method we find the existence of the relaxed skolam mean labeling of 5 - star graph with partition 3, 2 with a specific condition.

Relaxed Skolam Mean Labeling of 6 – Star Graphs with Partition 3, 3

Existence Relaxed skolam mean labeling for a 6 – star graph 3 G = K1,α K1,α K1,α K1,β K1,β K1,β 1 2 3 1 2 ∪ ∪ ∪ ∪ ∪ with partition 3,3 with a certain condition is the core topic of the following article. Trial and error method is used to find the existence of the relaxed skolam mean labeling of 6 - star graph with partition 3, 3 holding a specific condition.

The Component Diagnosability of General Networks

The processor failures in a multiprocessor system have a negative impact on its distributed computing efficiency. Because of the rapid expansion of multiprocessor systems, the importance of fault diagnosis is becoming increasingly prominent. The [Formula: see text]-component diagnosability of [Formula: see text], denoted by [Formula: see text], is the maximum number of nodes of the faulty set [Formula: see text] that is correctly identified in a system, and the number of components in [Formula: see text] is at least [Formula: see text]. In this paper, we determine the [Formula: see text]-component diagnosability of general networks under the PMC model and MM[Formula: see text] model. As applications, the component diagnosability is explored for some well-known networks, including complete cubic networks, hierarchical cubic networks, generalized exchanged hypercubes, dual-cube-like networks, hierarchical hypercubes, Cayley graphs generated by transposition trees (except star graphs), and DQcube as well. Furthermore, we provide some comparison results between the component diagnosability and other fault diagnosabilities.

Existence of solutions for a class of nonlinear boundary value problems on the hexasilinane graph

Chemical graph theory is a field of mathematics that studies ramifications of chemical network interactions. Using the concept of star graphs, several investigators have looked into the solutions to certain boundary value problems. Their choice to utilize star graphs was based on including a common point connected to other nodes. Our aim is to expand the range of the method by incorporating the graph of hexasilinane compound, which has a chemical formula $\mathrm{H}_{12} \mathrm{Si}_{6}$ H 12 Si 6 . In this paper, we examine the existence of solutions to fractional boundary value problems on such graphs, where the fractional derivative is in the Caputo sense. Finally, we include an example to support our significant findings.

b-Chromatic Sum and b-Continuity Property of Some Graphs

A b-coloring of a graph [Formula: see text] is a proper coloring of the vertices of [Formula: see text] such that there exist a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the largest integer [Formula: see text] such that [Formula: see text] has a b-coloring with [Formula: see text] colors. The b-chromatic sum of a graph [Formula: see text], denoted by [Formula: see text], is introduced and it is defined as the minimum of sum of colors [Formula: see text] of [Formula: see text] for any [Formula: see text] in a b-coloring of [Formula: see text] using [Formula: see text] colors. A graph [Formula: see text] is b-continuous, if it admits a b-coloring with [Formula: see text] colors, for every [Formula: see text]. In this paper, the [Formula: see text]-continuity property of corona of two cycles, corona of two star graphs and corona of two wheel graphs with unequal number of vertices is discussed. The b-continuity property of corona of any two graphs with same number of vertices is also discussed. Also, the b-continuity property of Mycielskian of complete graph, complete bipartite graph and paths are discussed. The b-chromatic sum of power graph of a path is also obtained.