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Article

Extremal Unicyclic Graphs for the Euler Sombor Index: Applications to Benzenoid Hydrocarbons and Drug Molecules

1
School of Mathematics and Computational Sciences, Huaihua University, Huaihua 418000, China
2
College of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(4), 249; https://doi.org/10.3390/axioms14040249
Submission received: 12 February 2025 / Revised: 11 March 2025 / Accepted: 25 March 2025 / Published: 26 March 2025

Abstract

:
With geometric significance, the Euler Sombor index of a graph Γ is defined as E P ( Γ ) = { u v } E ( Γ ) d ( u ) 2 + d ( v ) 2 + d ( u ) d ( v ) . It originates from the mathematical distance property and has been proven to have good chemical applications in octane isomers. In this paper, the minimum and maximum of the Euler Sombor index for unicyclic graphs with given girth, as well as the corresponding extremal graphs, are determined. As an application, the experimental values of this index for some benzenoid hydrocarbons and drug molecules were compared with the boiling point. Through regression analysis, it was further demonstrated that the Euler Sombor index has excellent predictability in the physicochemical properties of compounds.

1. Introduction

Topological indices are used to indicate and predict the physical and chemical properties of compounds [1,2]. It has widespread applications in chemistry, mathematics, and other fields and has become one of the most active research areas in chemical graph theory [3,4,5]. Many types of topological indices have been studied, among which the most extensively investigated and widely applied is undoubtedly vertex-degree-based topological indices (see [6,7,8]).
A vertex-degree-based topological index (VDB topological index) of Γ is as follows:
T I f ( Γ ) = { u v } E ( Γ ) f ( d ( u ) , d ( v ) ) ,
where f ( x , y ) is a pertinently chosen symmetric real function with x 1 and y 1 . In [9], motivated by geometrics, Gutman proposed a new concept for the Sombor index, which was defined by
S O ( Γ ) = { u v } E ( Γ ) d ( u ) 2 + d ( v ) 2 ,
where d ( u ) represents the degree of vertex u and { u v } takes over the edges of graph Γ .
Since its proposal, this index has attracted considerable attention and research interest. Gutman [9] investigated the extremum of the Sombor index for any tree T and connected graphs. Das et al. [10] provided lower and upper bounds for the Sombor index and established some relationships between the Sombor index and the first and second Zagreb indices. Furthermore, the extremal values of this index among all molecular trees were determined by Deng and Tang [11]. In [12], the authors deduced several upper and lower bounds of the Sombor index for some uniform hypergraphs. Chen and Zhu [13] established the upper and lower bounds of the Sombor index for unicyclic graphs, as well as the extrema of chemical unicyclic graphs with a fixed girth. Cruz and Rada [14] obtained the minimal values of (chemical) unicyclic and bicyclic graphs for the Sombor index and presented an upper bound on this index for such graphs. In 2024, Das [15] completely solved these open problems and characterized the extremal graphs. In addition, the authors [16,17,18] also studied the relevant chemical properties and applications of the Sombor index. For more research on this index, refer to review paper [19,20,21,22].
Recently, a new type of Sombor index, i.e, the Euler Sombor index was proposed by Gutman [23] and Tang et al. [24], which is formulated as
E P ( Γ ) = { u v } E ( Γ ) d ( u ) 2 + d ( v ) 2 + d ( u ) d ( v ) = f ( d ( u ) , d ( v ) ) .
Both articles explained that this index originates from an approximate expression of the circumference of an ellipse and is therefore a geometry-based invariant. Meanwhile, Gutman [23] determined some actual relations between the Euler Sombor index and the Sombor index. In [24], the authors determined the extremal values for this Euler Sombor index among all (molecular) trees and characterized the corresponding extremum graphs. More importantly, by studying the relevant chemical properties of octane isomers, such as the boiling point, acentric factor, entropy, SNar, HNar, etc., it has been demonstrated that the Euler Sombor index has excellent predictive applicability for physicochemical properties.
The purpose of our work was to investigate the extremum Euler Sombor index of unicyclic graphs with given girth l, with a focus on its applications for chemical and physical properties. In Section 2 and Section 3, the minimum and maximum Euler Sombor index of unicyclic graphs with given girth are identified and the corresponding extremum graphs are characterized. In Section 4, as an application of the Euler Sombor index, the experimental values of this index for some benzenoid hydrocarbons and drug molecules are compared with the boiling point. Through regression analysis, it is further demonstrated that the Euler Sombor index has excellent predictability of the physicochemical properties of compounds.
Now, we review some definitions and terms that will be used in the following text. In this article, we consider graphs as connected, simple, and undirected. A graph Γ = ( V , E ) , where the vertex set and edge set are denoted by V ( Γ ) and E ( Γ ) , respectively. The degree of vertex v is denoted by d Γ ( v ) or d ( v ) . In particular, a vertex v is called pendant if d ( v ) = 1 . The neighbors of vertex v, denoted by N Γ ( v ) or N ( v ) , is the collection of vertices adjacent to v. An edge { u v } is called pendant if d ( u ) = 1 or d ( v ) = 1 . Γ { u v } represents the subgraph from Γ by deleting the edge { u v } . Meanwhile, Γ + { u v } represents the graph by joining a new edge { u v } from Γ . Recall that a graph Γ is unicyclic if | E ( Γ ) | = | V ( Γ ) | . Let U n , l represent the set of unicyclic graphs with given girth l and an order of n. Clearly, if l = n , then U n , n = C n , where C n be the cycle of n vertices.

2. The Minimal Euler Sombor Index in U n , l

Lemma 1.
Let Γ 1 U n , l be as shown in Figure 1, where d ( u ) 3 , r s 1 and Γ is a subgraph of Γ 1 . Suppose that Γ 2 = Γ 1 u u 1 + v r u 1 . Then, E P ( Γ 2 ) < E P ( Γ 1 ) .
Proof. 
Assume that N Γ 1 ( u ) = { u 1 , v 1 , x 1 , x 2 , x t } and N Γ ( u ) = { x 1 , x 2 , x t } , where t 1 . Thus, we have d Γ 1 ( u ) = t + 2 . Now, three possibilities occur according to the values of d Γ 1 ( u 1 ) and d Γ 1 ( v 1 ) .
Case 1. If d Γ 1 ( u 1 ) = d Γ 1 ( v 1 ) = 1 . Since t 1 , combining the structures of Γ 1 and Γ 2 , we obtain
E P ( Γ 1 ) E P ( Γ 2 ) = Σ i = 1 t f ( t + 2 , d Γ ( x i ) ) + 2 f ( t + 2 , 1 ) Σ i = 1 t f ( t + 1 , d Γ ( x i ) ) + f ( t + 1 , 2 ) + f ( 2 , 1 ) Σ i = 1 t ( t + 2 ) 2 + x i 2 + x i ( t + 2 ) ( t + 1 ) 2 + x i 2 + x i ( t + 1 ) + 2 t 2 + 5 t + 7 t 2 + 4 t + 7 7 2 t 2 + 5 t + 7 t 2 + 4 t + 7 7 > 13 7 > 0 .
Case 2. If d Γ 1 ( v 1 ) = 2 ,   d Γ 1 ( u 1 ) = 1 . Similar to Case 1, it can be concluded that
E P ( Γ 1 ) E P ( Γ 2 ) = Σ i = 1 t f ( t + 2 , d ( x i ) ) f ( t + 1 , d ( x i ) ) + f ( t + 2 , 2 ) f ( t + 1 , 2 ) + f ( t + 2 , 1 ) f ( 2 , 2 ) > t 2 + 6 t + 10 t 2 + 4 t + 7 + t 2 + 5 t + 7 12 > t 2 + 6 t + 10 12 17 12 > 0 .
Case 3. If d Γ 1 ( v 1 ) = d Γ 1 ( u 1 ) = 2 , it follows that
E P ( Γ 1 ) E P ( Γ 2 ) = Σ i = 1 t f ( t + 2 , d Γ ( x i ) ) f ( t + 1 , d Γ ( x i ) ) + 2 f ( t + 2 , 2 ) + f ( 2 , 1 ) f ( t + 1 , 2 ) 2 f ( 2 , 2 ) > 2 t 2 + 6 t + 12 + 7 t 2 + 4 t + 7 2 12 > t 2 + 6 t + 12 + 7 2 12 19 + 7 2 12 > 0 .
Therefore, based on the above three cases, we can conclude that E P ( Γ 1 ) E P ( Γ 2 ) > 0 . The lemma is proven. □
Lemma 2.
Let Γ 3 U n , l be as shown in Figure 2, where u , v E ( C l ) , d ( u ) = d ( v ) = 3 and r s 1 . Assume that Γ 4 = Γ 3 u u 1 + v r u 1 . Then, we have E P ( Γ 4 ) < E P ( Γ 3 ) .
Proof. 
We assume that N Γ 3 ( u ) = { u 1 , x 1 , x 2 } and N Γ 4 ( v ) = { v 1 , y 1 , y 2 } . If u v C l , then x 2 = v , y 2 = u . We can use this for the following three cases.
Case 1. d Γ 3 ( u 1 ) = d Γ 3 ( v 1 ) = 1 . Since 3 2 + x i 2 + 3 x i 2 2 + x i 2 + 2 x i > 0 for i = 1 and 2, we deduce that
E P ( Γ 3 ) E P ( Γ 4 ) = Σ i = 1 2 f ( 3 , d Γ 3 ( x i ) ) + 2 f ( 3 , 1 ) Σ i = 1 2 f ( 2 , d Γ 3 ( x i ) ) + f ( 3 , 2 ) + f ( 2 , 1 ) Σ i = 1 2 3 2 + x i 2 + 3 x i 2 2 + x i 2 + 2 x i + 2 13 19 7 > 2 13 19 7 > 0 .
Case 2. d Γ 3 ( u 1 ) = 2 ,   d Γ 3 ( v 1 ) = 1 . It follows that
E P ( Γ 3 ) E P ( Γ 4 ) = Σ i = 1 2 f ( 3 , d Γ 3 ( x i ) ) + f ( 3 , 1 ) + f ( 2 , 1 ) Σ i = 1 2 f ( 2 , d Γ 3 ( x i ) ) + f ( 2 , 2 ) + f ( 2 , 1 ) Σ i = 1 2 3 2 + x i 2 + 3 x i 2 2 + x i 2 + 2 x i + 13 12 > 13 12 > 0 .
Case 3. d Γ 3 ( u 1 ) = d Γ 3 ( v 1 ) = 2 . If u v C l , we immediately have
E P ( Γ 3 ) E P ( Γ 4 ) = f ( 3 , d Γ 3 ( x 1 ) ) + f ( 3 , 3 ) + f ( 3 , 2 ) + f ( 2 , 1 ) f ( 2 , d Γ 3 ( x 1 ) ) + f ( 3 , 2 ) + 2 f ( 2 , 2 ) ( 3 2 + x 1 2 + 3 x 1 2 2 + x i 2 + 2 x 1 ) + 27 + 13 + 7 19 2 12 > 27 + 13 + 7 19 2 12 > 0 .
Similarly, if u v C l , then we have
E P ( Γ 3 ) E P ( Γ 4 ) = Σ i = 1 2 f ( 3 , d Γ 3 ( x i ) ) + f ( 3 , 2 ) + f ( 2 , 1 ) Σ i = 1 2 f ( 2 , d Γ 3 ( x i ) ) + 2 f ( 2 , 2 ) Σ i = 1 2 3 2 + x i 2 + 3 x i 2 2 + x i 2 + 2 x i + 19 + 7 2 12 > 19 + 7 2 12 > 0 .
Consequently, the above discussion implies that E P ( Γ 3 ) E P ( Γ 4 ) > 0 . This completes the proof. □
Let M n , l U n , l , and M n , l is obtained by attaching the path P = v 1 v 2 v n l to the cycle of C l via the vertex v C l . Now, we present the main results.
Theorem 1.
Let Γ U n , l . Then,
E P ( Γ ) 2 3 ( n 4 ) + 3 19 + 7 , if 3 l n 2 , 2 3 ( n 3 ) + 2 19 + 13 , if l = n 1 , 2 3 n , if l = n .
equality holds if and only if Γ M n , l .
Proof. 
Firstly, for l = n and l = n 1 , by direct calculation, we have
E P ( Γ ) = E P ( M n , n ) = E P ( C n ) = n 2 2 + 2 2 + 2 2 = 2 3 n ,
and
E P ( Γ ) = E P ( M n , n 1 ) = ( n 3 ) 2 2 + 2 2 + 2 · 2 + 2 2 2 + 3 2 + 2 · 3 + 1 2 + 3 2 + 3 = 2 3 ( n 3 ) + 2 19 + 13 .
Next, we consider the case 3 l n 2 . Suppose C = v 1 v 2 v l v 1 is the unique cycle in Γ . Consequently, Γ C n , and there is at least a vertex v i , such that d ( v i ) 3 .
If there are at least two vertices in { v 1 , v 2 , , v l } whose degrees are greater than or equal to 3, then Lemmas 1 and 2 can be applied to obtain
E P ( Γ ) > E P ( M n , l ) = 2 3 ( n 4 ) + 3 19 + 7 .
If there is exactly one vertex in { v 1 , v 2 , , v l } whose degree is greater than or equal to 3, without loss of generality, let v i . If d ( v i ) 4 , according to Lemma 1, we obtain
E P ( Γ ) > E P ( M n , l ) = 2 3 ( n 4 ) + 3 19 + 7 .
If d ( v i ) = 3 , by Lemma 1, we deduce that
E P ( Γ ) E P ( M n , l ) = 2 3 ( n 4 ) + 3 19 + 7 ,
and equality holds when Γ M n , l . Therefore, from the above discussion, the theorem holds. □

3. The Maximal Euler Sombor Index in U n , l

Let H n , l U n , l be a unicylic graph obtained by attaching n l pendant edges to the vertex u V ( C l ) . In particular, H n , 3 is denoted as S n + e .
Lemma 3.
Let f ( s , t ) = ( s + t + 2 ) 2 + s + t + 3 ) ( s + 2 ) 2 + s + 3 ( t + 2 ) 2 + t + 3 + 12 , where s , t 1 . Then, f m i n ( s , t ) = f ( 1 , 1 ) > 0 .
Proof. 
Since 1 1 + 3 4 ( x + 2 + 1 2 ) 2 is increasing with x 1 , we have
f ( s , t ) s = ( s + t + 2 ) + 1 2 ( s + t + 2 ) 2 + ( s + t + 2 ) + 1 ( s + 2 ) + 1 2 ( s + 2 ) 2 + ( s + 2 ) + 1 = 1 1 + 3 4 ( s + t + 2 + 1 2 ) 2 1 1 + 3 4 ( s + 2 + 1 2 ) 2 > 0 .
Similarly, it can be concluded that
f ( s , t ) t = ( s + t + 2 ) + 1 2 ( s + t + 2 ) 2 + ( s + t + 2 ) + 1 ( t + 2 ) + 1 2 ( t + 2 ) 2 + ( t + 2 ) + 1 = 1 1 + 3 4 ( s + t + 2 + 1 2 ) 2 1 1 + 3 4 ( t + 2 + 1 2 ) 2 > 0 .
Consequently, we obtain f m i n ( s , t ) = f ( 1 , 1 ) = 21 + 12 2 13 > 0 , and Lemma 3 is finished. □
Lemma 4.
Let f ( s , t ) = 2 ( s + t + 2 ) 2 + s + t + 3 + 2 ( s + t + 2 ) 2 + 2 ( s + t ) 2 + 8 + 2 12 ( s + 2 ) 2 + s + 3 ( t + 2 ) 2 + t + 3 2 ( s + 2 ) 2 + 2 s + 8 2 ( t + 2 ) 2 + 2 t + 8 , where s , t 1 . Then, f m i n ( s , t ) = f ( 1 , 1 ) > 0 .
Proof. 
Because 1 1 + 3 4 ( x + 2 + 1 2 ) 2 and 2 ( x + 2 ) + 2 are increasing and decreasing with x 1 , respectively, we deduce that
f ( s , t ) s = 2 ( s + t + 2 ) + 1 ( s + t + 2 ) 2 + ( s + t + 2 ) + 1 + 2 ( s + t + 2 ) + 2 ( s + t + 2 ) 2 + 2 ( s + t + 2 ) + 4 ( s + 2 ) + 1 2 ( s + 2 ) 2 + ( s + 2 ) + 1 2 ( s + 2 ) + 2 ( s + 2 ) 2 + 2 ( s + 2 ) + 4 > 1 1 + 3 4 ( s + t + 2 + 1 2 ) 2 1 1 + 3 4 ( s + 2 + 1 2 ) 2 + 2 ( s + 2 ) + 2 2 ( s + t + 2 ) + 2 > 0 .
Similarly, we obtain
f ( s , t ) t > 1 1 + 3 4 ( s + t + 2 + 1 2 ) 2 1 1 + 3 4 ( t + 2 + 1 2 ) 2 + 2 ( t + 2 ) + 2 2 ( s + t + 2 ) + 2 > 0 .
Therefore, we have f m i n ( s , t ) = f ( 1 , 1 ) = 2 21 + 2 28 + 12 4 19 2 13 > 0 , and the proof is completed. □
Lemma 5.
Let Γ 1 U n , l , and T is a component of Γ 1 C l , where { u v } T and u V ( C l ) . If | T | 2 , then the edge { u v } is contracted to u, and joining a pendant edge { u w } is denoted it by Γ 2 . Thus, E P ( Γ 1 ) < E P ( Γ 2 ) .
Proof. 
Suppose that N Γ 1 ( u ) = { v , x 1 , x 2 , , x t } and N Γ 1 ( v ) = { u , y 1 , y 2 , , y s } , where s , t 1 . Since f ( x , y ) = x 2 + y 2 + x y increases with x 1 , we have
E P ( Γ 2 ) E P ( Γ 1 ) = Σ i = 1 t f ( d ( u ) + d ( v ) 1 , d ( x i ) ) f ( d ( u ) , d ( x i ) ) + Σ i = 1 s f ( d ( u ) + d ( v ) 1 , d ( x i ) ) f ( d ( v ) , d ( x i ) ) + f ( d ( u ) + d ( v ) 1 , 1 ) f ( d ( u ) , d ( v ) ) ( t + s + 1 ) 2 + t + s + 2 ( t + 1 ) 2 + ( s + 1 ) 2 + ( t + 1 ) ( s + 1 ) t 2 + s 2 + 3 t + 3 s + 3 + 2 t s t 2 + s 2 + 3 t + 3 s + 3 + t s > 0 .
Therefore, the lemma holds true. □
Lemma 6.
Let Γ 3 U n , l ; { u u 1 , u u 2 , , u u s } and { v v 1 , v v 2 , , v v t } are the pendant edges joining to u and v, where u , v V ( C l ) and s , t 1 . Assume that Γ 4 = Γ 3 { v v 1 , v v 2 , , v v t } + { u v 1 , u v 2 , , u v t } ; then, E P ( Γ 3 ) < E P ( Γ 4 ) .
Proof. 
Assume that N Γ 3 ( u ) = { x 1 , x 2 , u 1 , u 2 , , u s } and N Γ 3 ( v ) = { y 1 , y 2 , v 1 , v 2 , , v t } . If u v E ( Γ 3 ) , then u = y 1 , v = x 1 . We consider two cases according to whether { u v } E ( C l ) or not.
Case 1. { u v } E ( C l ) . Since f ( d ( u ) + d ( v ) 2 , d ( x 2 ) ) f ( d ( u ) , d ( x 2 ) ) > 0 , we have
E P ( Γ 4 ) E P ( Γ 3 ) = Σ i = 1 s f ( d ( u ) + d ( v ) 2 , d ( x i ) ) f ( d ( u ) , d ( x i ) ) + Σ i = 1 t f ( d ( u ) + d ( v ) 2 , d ( x i ) ) f ( d ( v ) , d ( x i ) ) + f ( d ( u ) + d ( v ) 2 , 2 ) f ( d ( u ) , d ( v ) ) + f ( d ( u ) + d ( v ) 2 , d ( x 2 ) ) f ( d ( u ) , d ( x 2 ) ) + f ( 2 , d ( y 2 ) ) f ( d ( v ) , d ( y 2 ) ) > 2 ( s + t + 2 ) 2 + s + t + 3 + ( s + t + 2 ) 2 + 2 ( s + t ) 2 + 8 + 12 ( s + 2 ) 2 + s + 3 ( t + 2 ) 2 + t + 3 ( s + 2 ) 2 + ( t + 2 ) 2 + ( s + 2 ) ( t + 2 ) ( t + 2 ) 2 + s 2 + 2 t + 8 .
Through direct calculations, it can be concluded that
( s + t + 2 ) 2 + s + t + 3 > ( t + 2 ) 2 + s 2 + 2 t + 8 ,
( s + t + 2 ) 2 + 2 ( s + t ) 2 + 8 > ( s + 2 ) 2 + ( t + 2 ) 2 + ( s + 2 ) ( t + 2 ) .
Consequently, based on the above two equations and Lemma 3, we have
E P ( Γ 4 ) E P ( Γ 3 ) > ( s + t + 2 ) 2 + s + t + 3 + 12 ( s + 2 ) 2 + s + 3 ( t + 2 ) 2 + t + 3 > 0 .
Case 2. { u v } E ( C l ) . Note that N Γ 3 ( u ) = { x 1 , x 2 , u 1 , u 2 , , u s } and N Γ 3 ( v ) = { y 1 , y 2 , v 1 , v 2 , , v t } . From Case 1, for i = 1 , 2 , d ( x i ) = d ( y i ) = 2 . This together with Lemma 4 implies that
E P ( Γ 4 ) E P ( Γ 3 ) = Σ i = 1 s f ( d ( u ) + d ( v ) 2 , d ( x i ) ) f ( d ( u ) , d ( x i ) ) + Σ i = 1 t f ( d ( u ) + d ( v ) 2 , d ( x i ) ) f ( d ( v ) , d ( x i ) ) + 2 f ( d ( u ) + d ( v ) 2 , 2 ) f ( d ( u ) , 2 ) + 2 f ( 2 , 2 ) f ( d ( v ) , 2 ) 2 ( s + t + 2 ) 2 + s + t + 3 + 2 ( s + t + 2 ) 2 + 2 ( s + t ) 2 + 8 + 2 12 ( s + 2 ) 2 + s + 3 ( t + 2 ) 2 + t + 3 2 ( s + 2 ) 2 + 2 s + 8 2 ( t + 2 ) 2 + 2 t + 8 > 0 .
Therefore, the proof is finished. □
With the help of the above two lemmas, we can deduce the maximal Euler Sombor index in U n , l .
Theorem 2.
Let Γ U n , l . Then,
E P ( Γ ) ( n 3 ) n 2 n + 1 + 2 n 2 + 3 + 12 ,
and equality occurs when Γ H n , l .
Proof. 
Let C = v 1 v 2 v l v 1 be the unique cycle in Γ . Then, there is at least one vertex v i that meets with d ( v i ) 3 . If there are at least two vertices in { v 1 , v 2 , , v l } whose degrees are greater than or equal to 3, then according to Lemmas 5 and 6, we deduce that
E P ( Γ ) < E P ( H n , l ) = ( n 3 ) n 2 n + 1 + 2 n 2 + 3 + 12 .
If there is exactly one vertex in { v 1 , v 2 , , v l } whose degrees are greater than or equal to 3, and Γ H n , l , then, by Lemma 5, we have
E P ( Γ ) < E P ( H n , l ) = ( n 3 ) n 2 n + 1 + 2 n 2 + 3 + 12 .
If Γ H n , l , we have
E P ( Γ ) = E P ( H n , l ) = ( n 3 ) n 2 n + 1 + 2 n 2 + 3 + 12 .
Therefore, considering the above cases, the theorem holds. □

4. Chemical Applications of the Euler Sombor Index

There are various types of topological indices, and they have grown rapidly in recent years. One important role of topological indices is to predict the physical and chemical properties of molecules, such as the melting point, boiling point, etc. However, current research on topological indices mostly focuses on mathematical calculations, and it is very important to calculate topological indices through experiments and compare them with chemical and physical properties. Recently, Tang et al. [24] determined the relationship between the Euler Sombor index and the structure performance for octane. As an important compound, the study of the index and properties of benzene hydrocarbons is also of great significance. Therefore, in this section, we fill this gap.
Firstly, some typical benzenoid hydrocarbons (BHs) were considered (Figure 3). The experimental values of the Euler Sombor index (EP), Sombor index (SO), Elliptic Sombor index (ESO), forgotten topological index (F), symmetric division degree index (SDD), first (M1) and second (M2) Zagreb indices, and sum connectivity index (SCI), as well as the boiling point (BP) of benzenoid hydrocarbons, are shown in Table 1. In particular, |V(BH)| represents the size of molecules (the number of vertices) in BHs.
To eliminate the influence of molecular size on the prediction model, we adopted the method of dividing the obtained index value with the number of vertices and then compared different graphs of BHs to obtain the true prediction potentiality. From Table 1, we constructed the scatter chart among BP and E P values of BHs, as shown in Figure 4. Furthermore, we obtained the correlation coefficient (R) of various indices through regression analysis, as shown in Table 2. Therefore, the above results imply that the Euler Sombor index has important applicability in predicting the boiling point of BHs.
We conducted the above-mentioned research on some drug designs and mainly examined the ten chemical drugs listed in Table 3, whose molecular structures can be viewed online and are thus omitted here. The experimental values of the Euler Sombor index (EP), Sombor index (SO), Elliptic Sombor index (ESO), and boiling point (BP) of some drugs are shown in Table 3.
Similar to the operation of BHs, we eliminated the influence of molecular size on the prediction model by dividing the obtained index value with the number of vertices. Therefore, we constructed the scatter chart among BP and E P values of these drug molecules, as shown in Figure 5. Moreover, we obtained the correlation coefficient (R) of three types of Sombor index through regression analysis, as shown in Table 4. Clearly, the Euler Sombor index has a good ability to distinguish drug molecules and can be widely used in compound identification and drugs design.

5. Conclusions

In this paper, the minimum and maximum of the Euler Sombor index for unicyclic graphs with given girth were determined and the corresponding extremal graphs were characterized. In addition, by experimental comparison, it was confirmed that the Euler Sombor index can describe the physicochemical properties of benzene hydrocarbons and some drug structures. Through regression analysis, Figure 4 and Figure 5 show that both of the correlation coefficients (R) of the Euler Sombor index ( E P ) for benzenoid hydrocarbons and drug molecules with boiling point (BP) are greater than 0.8. This indicates that this index has good predictive ability for the physicochemical properties of compounds, especially those containing benzene rings. Therefore, it is of great significance for the design of new drugs and the development of new chemical materials.

Author Contributions

Writing—original draft, Z.S.; Writing—review & editing, Z.T. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Hunan Province Natural Science Foundation (2025JJ70485).

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Graph of Γ 1 .
Figure 1. Graph of Γ 1 .
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Figure 2. Graph of Γ 3 .
Figure 2. Graph of Γ 3 .
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Figure 3. Molecular graphs of some BHs.
Figure 3. Molecular graphs of some BHs.
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Figure 4. Linear fitting of E P with BP for some BHs.
Figure 4. Linear fitting of E P with BP for some BHs.
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Figure 5. Linear fitting of E P with BP for some drug molecules.
Figure 5. Linear fitting of E P with BP for some drug molecules.
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Table 1. Experimental values of some existing indices for BHs with BP.
Table 1. Experimental values of some existing indices for BHs with BP.
BH | V ( BH ) | BP EP SO ESO F SDD M 1 M 2 SCI
BH11021843.41635.635165.4499017250576.014
BH21433865.99154.160252.41617040976917.408
BH31434066.04854.300263.01618050076907.394
BH41843188.56572.685362.0112809951021259.619
BH51842588.62272.825361.2972849381021249.605
BH61842988.50772.545362.7252467291021269.633
BH71844088.68072.965360.58329810851021239.591
BH820496104.21185.553437.664318108012015110.830
BH920493104.15385.413438.37928686212015210.844
BH1020497104.15385.413438.37928686212015210.844
BH1122547119.85798.421513.318356120113817712.041
BH1222542119.79998.281514.03232699513817812.055
BH1322535111.13991.210460.292370134412815911.830
BH1422536111.19791.350459.578424178012815811.816
BH1522531111.19791.350459.578402158412815811.816
BH1622519111.13991.210460.292424178012815911.830
BH1724590135.445111.149589.686366112815620413.266
BH1824592126.727103.938536.659411148414618613.055
BH1924596126.785104.078535.945466192614618513.041
BH2024594126.785104.078535.945466192714618513.041
BH2124595126.727103.938536.659392134914618613.055
Table 2. The correlation coefficient of some existing indices for BHs with BP.
Table 2. The correlation coefficient of some existing indices for BHs with BP.
Index EP SO ESO F SDD M 1 M 2 SCI
R0.90090.90120.90380.80040.72690.90070.89710.2924
Table 3. Experimental values of three types of Sombor index for drug molecules with BP.
Table 3. Experimental values of three types of Sombor index for drug molecules with BP.
Drug Molecules | V ( Γ ) | BP EP SO ESO
Carmustine12404.040.20833.415146.254
Convolutamydine B16496.775.51263.265328.024
Bunitrolol17414.270.56259.579286.141
Perfragilin A17413.578.84165.774338.283
Melatonin17512.873.33660.810290.423
Aspirin20321.452.64244.042207.560
Podophyllotoxin29597.9142.744117.694603.003
Deguelin31560.1147.033134.880734.123
Aminopterin32551.7141.805118.149571.435
Minocycline35563.3184.633155.027863.063
Table 4. The correlation coefficient of three types of Sombor index for drug molecules with BP.
Table 4. The correlation coefficient of three types of Sombor index for drug molecules with BP.
Index EP SO ESO
R0.83190.83130.7911
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Su, Z.; Tang, Z. Extremal Unicyclic Graphs for the Euler Sombor Index: Applications to Benzenoid Hydrocarbons and Drug Molecules. Axioms 2025, 14, 249. https://doi.org/10.3390/axioms14040249

AMA Style

Su Z, Tang Z. Extremal Unicyclic Graphs for the Euler Sombor Index: Applications to Benzenoid Hydrocarbons and Drug Molecules. Axioms. 2025; 14(4):249. https://doi.org/10.3390/axioms14040249

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Su, Zhenhua, and Zikai Tang. 2025. "Extremal Unicyclic Graphs for the Euler Sombor Index: Applications to Benzenoid Hydrocarbons and Drug Molecules" Axioms 14, no. 4: 249. https://doi.org/10.3390/axioms14040249

APA Style

Su, Z., & Tang, Z. (2025). Extremal Unicyclic Graphs for the Euler Sombor Index: Applications to Benzenoid Hydrocarbons and Drug Molecules. Axioms, 14(4), 249. https://doi.org/10.3390/axioms14040249

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