Preparatory Year Deanship, King Faisal University, 31982, Hofuf, Al Ahsa,Saudi Arabia.
Department of Mathematics, COMSATS University Islamabad Lahore Campus, Lahore,Pakistan.
Comb Chem High Throughput Screen. 2022;25(3):500-509. doi: 10.2174/1386207323666201209094514.
A fullerene graph is a mathematical model of a fullerene molecule. A fullerene molecule, or simply a fullerene, is a polyhedral molecule made entirely of carbon atoms other than graphite and diamond. Chemical graph theory is a combination of chemistry and graph theory, where theoretical graph concepts are used to study the physical properties of mathematically modeled chemical compounds. Graph labeling is a vital area of graph theory that has application not only within mathematics but also in computer science, coding theory, medicine, communication networking, chemistry, among other fields. For example, in chemistry, vertex labeling is being used in the constitution of valence isomers and transition labeling to study chemical reaction networks Method: In terms of graphs, vertices represent atoms while edges stand for bonds between the atoms. By tvs (tes) we mean the least positive integer for which a graph has a vertex (edge) irregular total labeling such that no two vertices (edges) have the same weights. A (3,6)-fullerene graph is a non-classical fullerene whose faces are triangles and hexagons Results: Here, we study the total vertex (edge) irregularity strength of an arbitrary disjoint union of (3,6)-fullerene graphs and provide their exact values.
The lower bound for tvs (tes) depends on the number of vertices. Minimum and maximum degree of a graph exist in literature, while to get different weights, one can use sufficiently large numbers, but it is of no interest. Here, by proving that the lower bound is the upper bound, we close the case for (3,6)-fullerene graphs.
富勒烯图是富勒烯分子的数学模型。富勒烯分子，或简称富勒烯，是一种完全由碳原子组成的多面体分子，不同于石墨和钻石。化学图论是化学和图论的结合，其中理论图概念被用于研究数学建模化合物的物理性质。图标记是图论的一个重要领域，不仅在数学中，而且在计算机科学、编码理论、医学、通信网络、化学等领域都有应用。例如，在化学中，顶点标记用于构成价异构体和过渡标记，以研究化学反应网络。方法：在图论中，顶点表示原子，边表示原子之间的键。通过 tvs(tes)，我们指的是一个最小的正整数，对于这个正整数，一个图具有一个顶点（边）不规则的总标记，使得没有两个顶点（边）具有相同的权重。(3,6)-富勒烯图是非经典富勒烯，其面是三角形和六边形。结果：在这里，我们研究了任意不交并的(3,6)-富勒烯图的总顶点（边）不规则强度，并给出了它们的确切值。结论：tvs(tes)的下界取决于顶点的数量。图的最小和最大度在文献中已经存在，而要获得不同的权重，可以使用足够大的数字，但这没有意义。在这里，通过证明下界是上界，我们解决了(3,6)-富勒烯图的情况。
