Speaker
Description
Arithmetical structures on graphs have recently attracted considerable attention due to their rich connections with combinatorics, algebra, and graph theory. In this work, we undertake a detailed study of arithmetical structures on fan graphs. For a finite and connected graph G, an arithmetical structure is defined as a pair (d, r) of positive integer vectors such that the vector r is primitive and satisfies the relation (diag(d) − A)r = 0, where A denotes the adjacency matrix of G. We analyze the combinatorial properties of arithmetical structures associated with fan graphs, including the characterization and construction of such structures. Particular emphasis is placed on understanding how the underlying structure of fan graph influences these arithmetical configurations. In addition, we discuss the arrow-star graph, a graph derived from the fan graph, along with its properties, and investigate its structural properties in relation to arithmetical structures.