Cyclomatic Number in Molecular Graph Analysis

• Cyclomatic number is the count of independent cycles in a graph, serving as a key measure of cyclic complexity and the number of rings in molecular structures.
• It is calculated as |E(G)| - |V(G)| + 1 for connected graphs, linking vertex degrees to the graph’s topological properties.
• In minimizing the diminished Sombor index, the cyclomatic number governs the optimal degree sequences, resulting in either 3-regular or bidegreed molecular graphs.

The cyclomatic number is a fundamental structural invariant in graph theory and chemical graph theory, measuring the number of independent cycles in a graph. In molecular graph applications—where the graph’s maximum vertex degree is typically at most 4—the cyclomatic number quantifies the degree of cyclic complexity, directly correlating with the number of rings present in the molecular structure. This parameter is central when optimizing degree-based indices such as the diminished Sombor index (DSO), which encode information about molecular topology, stability, and electronic properties.

1. Definition of the Diminished Sombor Index and Cyclomatic Number

The diminished Sombor index (DSO) for a graph $G$, introduced as a degree-sensitive topological invariant, is defined as

$$\mathrm{DSO}(G) = \sum_{uv \in E(G)} \frac{ \sqrt{d(u)^2 + d(v)^2} }{ d(u) + d(v) },$$

where $d(u)$ is the degree of vertex $u$ and $E(G)$ denotes the edge set.

The cyclomatic number $\ell$ is specified as the minimum number of edges whose removal yields an acyclic (tree) graph, given by

$\ell = |E(G)| - |V(G)| + 1$

for any connected graph $G$.

In the context of molecular graphs—connected graphs with $\Delta(G)\leq 4$ and typically $\delta(G)\geq 2$—$\ell$ counts the rings (independent cycles), providing an intrinsic measure of molecular topology.

2. Classification of Extremal Molecular Graphs Minimizing the DSO Index

The paper offers a precise classification of molecular graphs which achieve the minimum DSO index for given order $n\geq 2(\ell-1)\geq 4$ and cyclomatic number $\ell\geq 3$.

Case 1: $n = 2(\ell-1)$

• The extremal graph is 3-regular: every vertex has degree 3.
• This is the unique configuration achieving the minimum DSO index in this regime.

Case 2: $n > 2(\ell-1)$

• Extremal graphs are bidegreed, with minimum degree 2 and maximum degree 3.
• Edge partitioning:
  • $m_{2,2}(G) = n - 2\ell + 1$ (edges between degree-2 vertices)
  • $m_{2,3}(G) = 2$ (edges between degree-2 and degree-3 vertices, must be even)
  • $m_{3,3}(G) = 3\ell - 4$ (edges between degree-3 vertices)

These constraints rigidly determine the structure, ensuring only graphs with this edge degree composition can achieve the minimal DSO index in the specified parameter regime.

Regime	Structural Type	Edge Counts
$n=2(\ell-1)$	3-regular	all degrees $=3$
$n>2(\ell-1)$	$(2,3)$-bidegreed	see counts above

3. Lower Bound Formulation for the Minimal DSO Index

An explicit lower bound is given for the minimal DSO index in terms of $n$ and $\ell$:

$$\mathrm{DSO}(G) \geq \frac{n + \ell - 3}{\sqrt{2} + (2\sqrt{13})/5}$$

The proof leverages the characteristic structural properties of extremal graphs for $n\geq 2(\ell-1)$, ensuring that equality is achieved if and only if the aforementioned classification holds.

Further details in the proof:

• The denominator arises from evaluating the index for edge types $(2,2)$, $(2,3)$, $(3,3)$.
• The sum reflects contributions weighted by the number of edges for each degree pair.

4. Cyclomatic Number as a Determinant in Extremal Topology

The cyclomatic number $\ell$ directly controls the possible edge distributions and degree sequences in molecular graphs of given order $n$:

• For fixed $n$, higher $\ell$ forces more degree-3 vertices and increases $m_{3,3}(G)$, reducing $m_{2,2}(G)$.
• When $n=2(\ell-1)$, the only valid configuration is the fully 3-regular graph, maximizing cyclic structure for a given size.
• When $n > 2(\ell-1)$, the minimal index requires a precise proportion of degree-2 and degree-3 vertices, governed by $\ell$.

This sharp dependence means that the cyclomatic number effectively ‘locks in’ the spectrum of possible minimized DSO configurations for molecular graphs of fixed order.

5. Resolution of an Earlier Conjecture and Structural Insights

An earlier conjecture ([F. Movahedi et al.], as referenced in the paper) claimed that for fixed order and $\ell=3$, the minimal DSO index is realized by connecting two disjoint cycles with two edges to create a quadrangle. The actual extremal graphs, found in the present work, differ:

• For $n=4$, only the 3-regular graph is extremal.
• For larger $n$, only graphs with the partitioned edge counts above minimize the index; the “quadrangle plus cycles” construct corresponds to a specific instance of the $(2,3)$-bidegreed family, not a universal structure.

This clarifies that the conjecture holds only for a single value of $n$ and a particular construction, while the general minimizers involve more nuanced degree sequences and edge distributions determined by $n$ and $\ell$.

6. Broader Implications for Chemical Graph Theory

These results substantially advance the understanding of degree-based indices in chemical graph theory:

• The DSO index, unlike raw sums of degrees, penalizes high degrees—hence, minimizing the index requires balancing the degree sequence while producing sufficient cycles.
• For molecular modeling, graphs minimizing the DSO index at fixed $n$ and $\ell$ have maximal symmetry when $n=2(\ell-1)$ (3-regular), and a sharply specified bidegree structure otherwise.
• These characterizations enable practitioners to identify, generate, and analyze molecular graphs with optimal properties for DSO-related chemical descriptors.

In summary, the cyclomatic number is the controlling invariant for the minimal DSO index in molecular graphs of given order, dictating not only the possible degree and edge distributions but also uniquely specifying the extremal graph families. The deep interplay between $\ell$, $n$, and degree statistics is now precisely characterized for this class of indices, resolving prior conjectural gaps and guiding future studies in graph-theoretic modeling of molecular topology (Alotaibi et al., 15 Sep 2025).