Aug 2026· Match-communications in Mathematical and in Computer Chemistry· 0 citations· 7 references
Abstract
In this research article, we replace the integer degree n = deg(v) of a vertex by its q(t)-analogue [n]q(t) and by an h(t)-analogue ⟨n⟩h(t) obtained from the h(t)-derivative of x n at x = 1, and we show that every classical degree-based index TI(G) = ∑︁uv∈E(G) F(deg u, deg v) admits two functional companions TIq(t)(G) and TIh(t)(G) that recover TI(G) exactly in the limit t → 0. We establish convergence, monotonicity, boundedness and a functional congruence theorem for these deformed indices, specialise the construction to the first and second Zagreb indices, the Randi´c index, the ABC index, the GA index, the harmonic index and the Sombor index, and verify all theoretical claims numerically on path, cycle, star and fused-ring (naphthalene) molecular graphs.
We then give an explicit chemical application, showing how an edge-weighted extension of the same framework tracks a degreebased topological index continuously along the reaction coordinate of a Diels–Alder cycloaddition, recovering the classical reactant- and product-graph indices at the two endpoints.
The zero-divisor graph of a commutative ring, Γ(R), is a prominent subject in graph theory. Previous research has investigated the energy and topological indices of Γ(ℤₚ[x]/⟨x⁴⟩) for any prime p; however, errors were identified in the characteristic polynomial and the associated energy bounds. Recent work by Rather add...
Vira Hari Krisnawati, Ayunda Faizatul Musyarrofah, Ari Andari· Science and Technology Indon...· 0 citations
Topological indices can be used to characterize molecular topology. These are numerical measurements of a suggested molecule’s fundamental structural characteristics, derived from its molecular structure. This numerical value, which is derived from a chemical configuration, represents the important physical properties...
Jayavel Pooja, Venkatesan Muthukumaran, B. Rather et al.· Discrete Mathematics, Algori...· 0 citations
Let G=(V,E) be a simple connected graph, where V and E denote the vertex and edge sets, respectively. The first Zagreb index is defined as H1(G)=∑ν∈VdegG(ν)2. Meanwhile, the second Zagreb index is given by H2(G)=∑ν1ν2∈EdegG(ν1)degG(ν2), where degG(ν) represents the degree of vertex ν. Another notable degree-based invar...
W. Ali, M. N. Husin, M. Nadeem et al.· Malaysian journal of mathema...· 0 citations
Investigating the descendant tree T(Delta) of 3-groups G of maximal nilpotency-class, with root Delta=SmallGroup(27,3), we show for the first time that an entropy-value H(P_r)=sum P_r(G)|log(P_r(G))| can also be determined for the real probability-distribution P_r(G) of all terminal vertices outside of the mainline on...
Let G be a graph with edge set E(G). The degree of a vertex w in G is denoted by d(w). The first-inverse Nirmala (FIN) index and the atom-bond connectivity (ABC) index of the graph G are defined, respectively, as FIN(G)=∑uv∈E(G)(d(u))−1+(d(v))−1 and ABC(G)=∑uv∈E(G)(d(u))−1+(d(v))−1−2(d(u)d(v))−1. The problem of determi...
In chemical graph theory, a molecular structure is modeled as a connected graph where vertices represent atoms and edges represent chemical bonds. A pentagonal chain is an organic network composed of fused five-membered cycles. While previous research has examined single, double, triple, and quadruple pentagonal config...
Thukarama V., J. K· International journal of mul...· 0 citations
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