Approximation of Algebraic Curves in Function Spaces of Topological Sequences Connected to Specific Simple Graph Families

Algebraic curves and topological sequences play a crucial role in mathematics and graph theory, serving as a bridge between geometry, algebra, and number theory. They facilitate structural analysis in various applications, including chemistry, network analysis, and computer science. In this research, we introduce the concept of estimated algebraic curves and topological sequences S T , and we explore the development and analysis of estimated linear and exponential algebraic curves that emerge from topological sequences associated with collections of simple graphs. By analyzing various graph invariants and their corresponding sequences, we aim to estimate and model these sequences through algebraic curves, thereby shedding light on their structural dynamics and growth trends. Our study centers on some families of graphs, such as snake and pan graphs, from which we obtain closed‐form expressions and asymptotic approximations for their corresponding topological sequences. By constructing linear and exponential algebraic curves, we characterise the mathematical interactions regulating the evolution of these sequences and explain how the structural qualities of graphs affect their topological properties. These algebraic curves provide new approaches to comprehend complicated networks and help address graph‐theoretic problems in combinatorics, network research, and computational geometry.