On Unicyclic Graphs with Minimum Graovac–Ghorbani Index

: In discrete mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Chemical graph theory is concerned with non-trivial applications of graph theory to the solution of molecular problems. Its main goal is to use numerical invariants to reduce the topological structure of a molecule to a single number that characterizes its properties. Topological indices are numerical invariants associated with the chemical constitution, for the purpose of the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. They have found important application in predicting the behavior of chemical substances. The Graovac–Ghorbani ( ABC GG ) index is a topological descriptor that has improved predictive potential compared to analogous descriptors. It is used to model both the boiling point and melting point of molecules and is applied in the pharmaceutical industry. In the recent years, the number of publications on its mathematical properties has increased. The aim of this work is to partially solve an open problem, namely to find the structure of unicyclic graphs that minimize the ABC GG index. We characterize unicyclic graphs with even girth that minimize the ABC GG index, while we also present partial results for odd girths. As an auxiliary result, we compare the ABC GG indices of paths and cycles with an odd number of vertices.


Introduction
Let G be a simple connected undirected graph of order n = |V(G)| and size m = |E(G)|.The degree d(v) of a vertex v ∈ V(G) is the number of vertices adjacent to v. We write d G (v) if we want to emphasize the graph G in which the degree of a vertex v is considered.The distance d(u, v) between the vertices u and v is defined as the number of edges on the shortest path connecting u and v.In chemical graph theory, a graph is used to represent a molecule by considering the atoms as the vertices of the graph and the molecular bonds as the edges.
Molecular descriptors can be defined as mathematical representations of molecular properties generated by algorithms.The numerical values of molecular descriptors are used to quantitatively describe the physical and chemical information of molecules.Topological descriptors are molecular descriptors [1] that serve as a tool for the compact and effective description of structural formulas used to study and predict the structure-property correlation of organic compounds [2][3][4].Countless applications of topological indices have been reported, most of which are related to the study of medical and pharmacological issues.
The best known topological index seems to be the Randić connectivity index [5], which has numerous applications in chemistry and pharmacology, with a profound mathematical background.A quite successful descendant of the Randić index is the atom-bond connectivity (ABC(G)) index introduced by Estrada et al. in 1998 [6], as follows According to Furtula [7], the ABC index is one of the best degree-based molecular descriptors.
In 2010, Graovac and Ghorbani defined a new version of the atom-bond connectivity index, a distance-based topological descriptor known as the Graovac-Ghorbani (ABC GG ) index [8].It is defined as where n u is the number of vertices that are closer to the vertex u than to vertex v, and n v is the number of vertices that are closer to v than to u.It was pointed out in [7] that the ABC GG index provides significantly better correlations than the atom-bond connectivity index for certain physico-chemical properties.In recent years, the mathematical properties of the ABC GG index [9][10][11][12][13][14] have been intensively studied in the literature.Recently, a survey of the ABC GG index was presented in [15], which included a complete bibliography for future research.Its recentness and the current knowledge on the ABC GG index suggest that there are many opportunities for further research into its properties.For many types of graphs, extreme values of the ABC GG index are unknown.In 2013, Das et al. [16] found maximum values of the ABC GG index for unicyclic graphs, while the problem of finding minimum values for the same class of graphs has remained open.Throughout this paper, we investigate the properties of the ABC GG index in unicyclic graphs.We characterize unicyclic graphs with even girth that minimize the ABC GG index, while we present partial results for odd girth.As an auxiliary result, we compare the ABC GG indices of paths and cycles with an odd number of vertices.Our study is significant because it partially solves an open problem regarding the ABC GG index of unicyclic graphs using new mathematical results related to this quantity, which can be applied to other types of graphs.

Preliminaries
We present two lemmas related to summands in the definition (1) of the ABC GG index.
) is a strictly increasing function of x; (iii) For x ≥ 2 and y ≥ 2 it holds f (x, y) ≤ √ 1/2 and f is a decreasing function, i.e., ∀(x, y), (x ′ , y ′ ) ∈ N 2 it holds Proof.Let g : N 2 → R be a function defined by g(x, y) = x+y−2 xy .Then, f (x, y) = g(x, y), i.e., f is monotonic transformation of g (if g increases (decrases), then f increases (decreases)).Notice that g ( f ) is a symmetric function.It is easy to prove that claims (i) and (ii) hold for g, and consequently for ( f ).(iii) Let r, t ∈ N 0 and x, y ≥ 2. Then We conclude that g decreases if and only if y ≥ x + t.Therefore, f (x, y) ≥ f (x + t, y − t) if and only if y ≥ x + t.
Throughout this paper, for uv ∈ E(G) and the numbers n u and n v defined as in (1), f (n u , n v ) is called the gg-value of uv.
Lemma 2. For n ≥ 5, we have Proof.Cases n = 5 and n = 6 can be checked directly.Let n ≥ 7.Both sides of inequality (2) are increasing functions of n.For n ≥ 7, we have 2 n−3 n−2 ≥ 1.7889 and 1 2 + n−2 n−1 < and this completes the proof.

Main Results
Paths and cycles are fundamental concepts in graph theory, often considered as subgraphs of other graphs [17].A path graph P n is a graph whose vertices can be listed in the order 1, 2, . . ., n, so that the edges are {i, i + 1} for i = 1, . . ., n − 1.The cycle graph C n is derived from P n by connecting vertices 1 and n using an edge.A unicyclic graph G is a connected graph with exactly one cycle.This implies |E(G)| = n.We now compare the ABC GG indices for paths and cycles.

Graovac-Ghorbani Index of Paths and Cycles
In 2014, Rostami and Sohrabi-Haghighat found trees that minimize the ABC GG index.Theorem 1 ([18]).The path P n is the n−vertex tree with the minimum Graovac-Ghorbani index.
The Graovac-Ghorbani index of a path P n is given by the following formula: which can be written as From part (iv) of Lemma 1, we can observe that the gg-values of the edges in P n decrease as we move from pendant edges to the central one (ones).For an even n, the smallest gg-value is obtained for a single central edge and is equal to f (n/2, n/2) = 2 √ n−2 n , while for n odd, we have two central edges with the smallest gg-value f ((n − 1)/2, (n + 1)/2) = 2 n−2 n 2 −1 .In a cycle graph C n , all edges have the same gg-value.For n even, this is n−2 while for n odd, we have n−3 In [10], Dimitrov et al. investigated the ABC GG index of bipartite graphs.As an auxiliary result, they established that ABC GG (P n ) > ABC GG (C n ) for all even n ≥ 8, while for n ∈ {4, 6}, it holds ABC GG (P n ) < ABC GG (C n ).Here, we examine the case where n is odd.For this purpose, we need several auxiliary results.
Proof.Numerical calculations show that for n odd, 11 ≤ n ≤ 23 inequality holds.Let n ≥ 25.Then, n = 2t + 1, t ≥ 12 and inequality (5) can be written as It is easy to see that 2t+i i is a decreasing function of i.Therefore, all summands in (6) are decreasing and the last one is equal to 2t Notice that ( 10) is larger than 2t + 1 if and only if If we analyze the inequality (11) for each of the 12 possible pairs (k, l), we come to the conclusion that it holds for t ≥ 12.The results are summarized in Table 1 and the proof is complete.
Proof.As we mentioned above, for n even, n ≥ 4 inequalities were proven in [10].
Graovac-Ghorbani indices of P n and C n for some n are presented in Table 2.

Unicyclic Graphs
As we mentioned in the introduction, unicyclic graphs maximizing the ABC GG index were found in [16].To the best of our knowledge, the problem of minimizing the ABC GG index for unicyclic graphs has not been solved in general.By studying the ABC GG index of bipartite graphs.Dimitrov et al. [10] characterized unicyclic graphs with an even number of vertices and even girth in a non-explicit way that minimized the ABC GG index.By C ′ n we denote a unicyclic n−vertex graph consisting of a cycle C n−1 with a pendant vertex, and by C ′′ n we denote a graph with an odd number of vertices n comprised of two even cycles C n−1 and C 4 that have three common vertices and two common edges.

Theorem 3 ([10]
).Among all bipartite graphs on n ≥ 8 vertices, the minimum Graovac-Ghorbani index is attained by the cycle C n for even n, by C ′ n for odd n ≤ 15, and by C ′′ n for odd n ≥ 17.For n < 8, the graph that minimizes the Graovac-Ghorbani index is the path P n on n vertices.Furthermore, these are the unique graphs with these properties.
If we restrict ourselves to bipartite unicyclic graphs with an even number n of vertices, n ≥ 8, then a direct consequence of Theorem 3 states that for such n, the cycle C n is a unicyclic graph with even girth and minimal ABC GG index.
Pendant edge-moving transformation of a connected graph G. Let a ≥ b ≥ 1 and let G be a connected graph with an induced path (induced subgraph that is a path) P a+b+1 , in which only one internal vertex has a degree of at least 3. Let a be the number of vertices of P a+b+1 on one side of w, and b the number of vertices on the other side, see Figure 1.By moving a pendant vertex from the b−side of a path to its a−side, we perform a so-called pendant edge-moving transformation of G.
In [18], Rostami and Sohrabi-Haghighat proved the following lemma for trees.We generalize it to connected graphs.Then, H = (G 1 \ P a+b+1 ) + w and the pendant edge-moving transformation preserves the gg-values of the edges in H.We have Similarly, .
We obtain For s ∈ N, s ≥ 3, we denote by C(r 1 , r 2 , . . ., r s ) an n−vertex unicyclic graph consisting of a cycle C s , |V(C s )| = {v 1 , v 2 , . . ., v s } and paths P r i , r i ≥ 1, such that v i is an end vertex of P r i , i = 1, . . ., s.The vertices v 1 , . . ., v s are positioned clockwise on C s , see Figure 2. Theorem 4. Let G be a unicyclic graph with a cycle C s , s ≥ 3, V(C s ) = {v 1 , . . ., v s }, and let T r i be an r i −vertex tree in G containing v i , i = 1, . . ., s.Then ABC GG (G) ≥ ABC GG (C(r 1 , . . ., r s )).
Proof.We repeatedly apply a pendant-edge moving transformation to G; i.e., to each T r i , i = 1, . . ., s, we perform a sequence of pendant-edge moving transformations until we obtain a path P r i .These transformations preserve the unicyclic property of G, while Lemma 5 implies a reduction in the ABC GG index.
Due to Theorem 4, unicyclic graphs with minimal ABC GG index belong to the class of graphs C(r 1 , r 2 , . . ., r s ).Due to a different behavior, n−vertex unicyclic graphs of girth 3 are considered separately.
The calculations show that among all unicyclic graphs with 3 ≤ n ≤ 5 vertices, the graph C(n − 2, 1, 1) has the smallest ABC GG index.Theorem 5. Let n ≥ 6 and let G be an n−vertex unicyclic graph of girth 3. Then ABC GG (G) > ABC GG (C n ).
Next, we find the smallest gg-values of the edges of a cycle C s , s ≥ 4 in any unicyclic n−vertex graph G. Lemma 6.Let n ∈ N, n ≥ 4 and let G be a unicyclic graph with cycle C s , s ≥ 4. Then we have for each edge e = uv ∈ E(C s ) , for n even and s even, ( 12) , for n even and s odd, ( The equality is given if s = n, i.e., the edge e belongs to C n , or if s = n − 1, i.e., the edge e belongs to a cycle in C ′ n .The graphs C n and C ′ n are unique unicyclic graphs containing the maximum number of cycle edges with the smallest gg-values.

Proof.
Case 1: n and s are even.For each edge e = uv ∈ E(C s ), s ≥ 4 we have n u , n v ≥ 2, n u + n v = n and the largest value of the product n u n v is obtained for , which is a gg-value of an arbitrary edge of C n .
Case 2: n is even and s is odd.Then, s ≥ 5 and at least one vertex of G does not lie on a cycle C s .For e = uv ∈ E(C s ), we have n u , n v ≥ 2 and n u + n v = t ≤ n − 1, since there is at least one vertex that is equidistant from u and v. (For t = n − 1, such a vertex is unique t 2 −1 is a decreasing function of odd t ≥ 5 and reaches its minimum value for t = n − 1.
is a decreasing function of even t ≥ 4. It follows that 2 Let w be the vertex on a cycle for which d G (w) ≥ 3 (such a vertex exists since at least one vertex of G is not on a cycle).Then, there is a single edge f ∈ E(C s ) whose end vertices are equidistant from w. Since a tree attached to w exists, we conclude that the gg-value of f is greater than 2 n−3 n(n−2) and there exists at least one pendant edge in G having gg-value . Therefore, G contains at least 2 edges with a non-minimal gg-value.We conclude that the maximum number of cycle edges with the smallest gg-value is n − 2 and they belong to G = C ′ n .Case 3: n and s are odd.Then, s ≥ 5 and for a cycle edge e = uv, we have have the smallest gg-value.
In the following, we compare gg-values of edges in an arbitrary n−vertex tree with the smallest gg-values of cycle edges in an n−vertex unicyclic graph G.
and for n ≥ 5, it holds Proof.We have f (i, n − i) = n−2 i(n−i) and from Lemma 1 (iv), by taking t = 1 we obtain and the inequality ( 16) is proven.
To prove (17), notice that n−2 i(n−i) < which is a quadratic inequality of variable i.Its solutions are integers i from the interval , n 2 .Therefore, (17) holds.Similarly, we note that n−2 i(n−i) < 4 n−3 (n−1) 2 is equivalent to , n−1 2 and we have proven (18).
We are ready to characterize unicyclic graphs with even girth that minimize the Graovac-Ghorbani index.Theorem 6.For n ≥ 4, let G be an n−vertex unicyclic graph of even girth.Then Proof.Let us consider the case where n is even.The inequality (12) from Lemma 6 implies that the gg-value of each edge of a cycle in G is greater than or equal to the gg-value of C n , which is equal to 2 n √ n − 2.Moreover, inequality ( 16) from Lemma 7 implies that the gg-value of each edge of a tree in G (if any) is greater than or equal to the gg-value of C n .For n odd, the inequality (15) from Lemma 6 implies that the gg-value of each edge of a cycle in G is greater than or equal to the gg-value of a cycle edge in C ′ n , which is equal to 2 n−2 n 2 −1 .The inequality (16) from Lemma 7 implies that the gg-value of each edge of a tree in G (which exists) is greater than or equal to the gg-value of a cycle edge C ′ n .Since C ′ n contains a single pendant edge, we obtain and the inequality (19) is proven.
Proof.A simple calculation shows that the inequality holds for n = 5, 7. Let n ≥ 9.The Lemma 3 implies Therefore, For n ≥ 9, we have √ n(n − 3) ≥ 2n and for n ≥ 5, it holds Therefore, the inequality (20) holds and this completes the proof.
Corollary 1.Let n ≥ 4 and let G be an n−vertex unicyclic graph of girth s ≥ 4, s is even.Then Proof.The result follows directly from Theorem 6 and Lemma 8.
We continue our studies by examining unicyclic graphs G with odd girth s, where s ≥ 5. We say that the edge of a tree in G (if any) is gg-small if its gg-value n−2 i(n−i) satisfies the inequality (17) (if n is even) or the inequality (18) (if n is odd).Theorem 7. Let n ≥ 5 and let G be an n−vertex unicyclic graph of odd girth s ≥ 5 with zero gg-small edges.Then We conclude ABC GG (G) ≥ ABC GG (C n ).Case 2: n is even.Then, G contains at least one vertex that is not on the cycle.Consequently, it contains at least one pendant edge and for at least one cycle edge f = wz there are p ≥ 2 vertices equidistant from u and v.We have n w + n z = t = n − p.Note that p and t have the same parity and t ≤ n − 2 if t is even, while t ≤ n − 3 if t is odd.We omit the details and refer to Case 2 of Lemma 6 to conclude that f (n w , n z ) ≥ f (n/2 − 1, n/2 − 1) = 2 n−4 (n−2) 2 .The above considerations in combination with the inequality (13) and the reversed inequality in (17) result in Corollary 3. If a pendant edge-moving transformation of an unicyclic graph G with odd girth s ≥ 5 yields C(r 1 , . . ., r s ) with gg-small edges on two disjoint paths P r k and P r l , then ABC GG (G) > ABC GG (C n ).
Proof.Theorems 4, 8 and Lemma 9 give ABC GG (G) ≥ ABC GG (C(r 1 , . . ., r s )) Now, a single type of unicyclic graph with an odd girth remains to be investigated.This is a graph with gg-small edges whose pendant edge-moving transformation gives C(r 1 , . . . ,r s ) with gg-small edges on a single path.Numerical experiments indicate that many such graphs have an ABC GG index larger than ABC GG (C n ).However, at this moment, we are not able to provide a general proof of this conjecture, so we leave this for future research.

Conclusions
In this study, we investigated the Graovac-Ghorbani index for unicyclic graphs.As an auxiliary result, we first showed that for every n ≥ 8 the ABC GG index of the cycle C n is larger than the ABC GG index of the path P n .We characterized unicyclic graphs of even girth with the smallest ABC GG index using pendant edge-moving transformation.For unicyclic graphs with odd girth, we offer a conjecture based on an analysis of a large number of cases.

Figure 1 .Lemma 5 .
Figure 1.Pendant edge-moving transformation of a connected graph.Lemma 5. Let G be a connected n−vertex graph that allows the pendant edge-moving transformation, and let G 1 be the resulting graph.Then ABC GG (G 1 ) < ABC GG (G).

Proof.
Let a ≥ b ≥ 1 and let P a+b+1 be an induced path of G with a single internal vertex w, such that d G (w) ≥ 3.Then, w is a cut-vertex in both G and G 1 .Let H := (G \ P a+b+1 ) + w.

Conjecture 1 .
Let G be an n−vertex unicyclic graph with an odd girth s ≥ 5.Then ABC GG (G) ≥ ABC GG (C n ).

Funding:
The author received no funding for this work.
Now, we are ready to prove the main result.Theorem 2. For 4 ≤ n ≤ 7 it holds ABC GG (P n ) < ABC GG (C n ), while for n = 3 and for n ≥ 8 we have ABC GG 7, 9}.Let n ≥ 11, n odd.From Lemmas 3 and 4, it follows that

Table 2 .
Numerical values of Graovac-Ghorbani indices of P n and C n , 3 ≤ n ≤ 12.
Cycle C n is the unique graph in which all edges have the smallest gg-value.Case 4: n is odd and s is even.Then, s ≥ 4 and for any cycle edge in G, we have n u , n v ≥ 2 and n u