M-Polynomials and Zagreb Indices for i-Iterated Subdivided-Line Graphs

The i-iterated subdivided-line graph Γi(G) of a base graph G is the graph obtained from G by iteratively applying the subdividedline graph operation i times. The M-polynomial of a graph G is a bivariate polynomial that encodes the degree-based properties of G. In this paper, we present a general formula expressing the Mpolynomial of Γi (G) in terms of the degrees of the vertices of the base graph G. We compute the First and Second Zagreb indices from the M-polynomial of i-iterated subdivided-line graphs Γi (G) when G belongs to several graph classes, such as wheel, ladder, ∆- regular, cycle, and tadpole graphs. The obtained results generalize those of Ranjini et al. (2011). Additionally, we analyse the impact of the vertex of maximum degree on the value of the First and Second Zagreb indices, providing asymptotic upper bounds for i-iterated subdivided-line graphs of general graphs.