Next Article in Journal
PPVC: Towards a Personalized Local Differential Privacy-Preserving Scheme for V2G Charging Networks
Previous Article in Journal
Finite-Time Anti-Saturated Formation Tracking Control of Multiple Unmanned Aerial Vehicles: A Performance Tuning Way
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Ad-Hoc Lanzhou Index

1
Department of Mathematics, College of Science, University of Ha’il, Ha’il P.O. Box 2240, Saudi Arabia
2
Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK
3
Faculty of Information Studies, 8000 Novo Mesto, Slovenia
4
Donát Bánki Faculty of Mechanical and Safety Engineering, Óbuda University, H-1034 Budapest, Hungary
*
Author to whom correspondence should be addressed.
Mathematics 2023, 11(20), 4256; https://doi.org/10.3390/math11204256
Submission received: 25 September 2023 / Revised: 6 October 2023 / Accepted: 9 October 2023 / Published: 11 October 2023

Abstract

:
This paper initiates the study of the mathematical aspects of the ad-hoc Lanzhou index. If G is a graph with the vertex set { x 1 , , x n } , then the ad-hoc Lanzhou index of G is defined by L z ˜ ( G ) = i = 1 n d i ( n 1 d i ) 2 , where d i represents the degree of the vertex x i . Several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all connected ξ -cyclic graphs of a fixed order, (ii) the set of all connected molecular ξ -cyclic graphs of a fixed order, (iii) the set of all graphs of a fixed order, and (iv) the set of all connected molecular graphs of a fixed order.

1. Introduction

Graph invariants are regarded as the properties of a graph that the graph isomorphism preserves [1]. Real-valued graph invariants are often known as topological indices [2]. We mention [2,3,4,5] as sources for terminology and notations related to (chemical) graph theory.
One of the most extensively researched topological indices is the first Zagreb index, which originally appeared in [6]. For a graph G, its first Zagreb index is often represented by M 1 ( G ) and is defined (for example see [7]) as
M 1 ( G ) = t V ( G ) d t 2 = r s E ( G ) d r + d s ,
where d t represents the degree of the vertex t in G and E ( G ) is the set of edges of G. The forgotten topological index [8] (which is sometimes referred to as the F-index, see also [9]) is another index that first appeared in [6]. The F-index of a graph G is represented by F ( G ) and is defined [8] as follows:
F ( G ) = t V ( G ) d t 3 = r s E ( G ) d r 2 + d s 2 .
Vukičević et al. [10] studied (chemically as well as mathematically) the following linear combination of the indices M 1 ( G ) and F ( G ) for a graph of order n and referred it to as the Lanzhou index:
L z ( G ) = ( n 1 ) M 1 ( G ) F ( G ) .
The Lanzhou index can be rewritten as
L z ( G ) = t V ( G ) d t 2 d t ¯ ,
where G ¯ is the complement of G and d t ¯ represent the degree of t in G ¯ . The refs. [11,12,13,14,15] provide some recent extremal results regarding the Lanzhou index.
If T I ( G ) is a topological index of a graph G, then T I ( L ( G ) ) is known as its reformulated version [16,17]. Here, L ( G ) means the line graph of G. Motivated by the concept of reformulated topological indices [16,17], we consider ad-hoc topological indices as follows: if T I ( G ) is a topological index of a graph G, then we call T I ( G ¯ ) as the ad-hoc version of T I ( G ) . Thus, applying the idea of ad-hoc topological indices to the Lanzhou index gives the ad-hoc Lanzhou index, represented by L z ˜ . The ad-hoc Lanzhou index [10], for a graph G, is defined as
L z ˜ ( G ) = t V ( G ) d t d t ¯ 2 = L z ( G ¯ ) .
Here, we mention that the Lanzhou index of the complement of G was also studied in [18]. If | V ( G ) | = n , then d t ¯ = n d t 1 and thus
L z ( G ) = t V ( G ) d t 2 n d t 1 and L z ˜ ( G ) = t V ( G ) d t n d t 1 2 .
The ad-hoc Lanzhou index was examined in [10] for predicting the octanol–water partition coefficient of nonane isomers, and it was found that this index performs better than both the well-known first Zagreb index and the F-index.
A graph with n vertices is called an n-order graph. Molecular graphs are those with a maximum degree of at most 4. A connected ξ -cyclic graph of order n is a connected n-order graph with ξ + n 1 edges. For ξ = 0 , 1 , 2 , and 3, a connected ξ -cyclic graph is also known as a tree, connected unicyclic graph, connected bicyclic graph, and connected tricyclic graph, respectively.
In this paper, several identities for the ad-hoc Lanzhou index, involving some existing topological indices, are established. The problems of finding graphs with the extremum values of the ad-hoc Lanzhou index from the following sets of graphs are also attacked: (i) the set of all n-order connected ξ -cyclic graphs (with a particular emphasis on unicyclic graphs, trees, and bicyclic as well as tricyclic graphs), (ii) the set of all n-order connected molecular ξ -cyclic graphs, (iii) the set of all n-order graphs, and (iv) the set of all n-order connected molecular graphs.

2. Identities

For a graph G, its forgotten topological coindex (or simply the F-coindex) is represented by F ¯ and is defined [19,20] by
F ¯ ( G ) = s t E ( G ) d s 2 + d t 2 .
Actually, the F-coindex is equal to the Lanzhou index for every graph; see, for example, [21]. Generally, if s t E ( G ) f ( g ( s ) , g ( t ) ) is a topological index of a graph G then the corresponding coindex is defined as s t E ( G ) f ( g ( s ) , g ( t ) ) , where g may be the degree, the eccentricity, or any other (real-valued) parameter defined on the vertices of G and where f is a real-valued symmetric function. Note that the Lanzhou index can be rewritten as
L z ( G ) = s t E ( G ) d s ( G ) d s ( G ¯ ) + d t ( G ) d t ( G ¯ ) ,
where d s ( G ) and d s ( G ¯ ) indicate the degrees of the vertex s V ( G ) in G and G ¯ , respectively. When there is no chance of confusion, we drop “ ( G ) ” from the notation d s ( G ) . Applying the definition of a coindex to the Lanzhou index yields the Lanzhou coindex L z ¯ :
L z ¯ ( G ) = s t E ( G ) d s ( G ) d s ( G ¯ ) + d t ( G ) d t ( G ¯ ) .
Consequently, we have
L z ¯ ( G ) = s t E ( G ¯ ) d s ( G ¯ ) d s ( G ) + d t ( G ¯ ) d t ( G ) = L z ( G ¯ ) .
The following result is immediate from the above.
Observation 1.
For every graph G, its Lanzhou coindex is equal to the Lanzhou index of G ¯ (which is termed as the ad-hoc Lanzhou index of G), which is equal to the F-coindex of G ¯ ; that is,
L z ¯ ( G ) = L z ( G ¯ ) = L z ˜ ( G ) = F ¯ ( G ¯ ) .
Because of (2), the identity given in the following proposition is already known (see Equation (3.6) in [20]); however, here we provide its more simple proof.
Proposition 1.
For any graph G with size m and order n, the following identity holds:
L z ˜ ( G ) = 2 ( n 1 ) 2 m + F ( G ) 2 ( n 1 ) M 1 ( G ) .
Proof. 
Note that the formula for L z ˜ can be rewritten as
L z ˜ ( G ) = r s E ( G ) n d r 1 2 + n d s 1 2 .
Expanding the squared terms in (3) and then making use of the definitions of F and M 1 , we obtain the desired identity. □
By Proposition 1, every upper bound on M 1 provides a lower bound on L z ˜ and every lower bound on M 1 gives an upper bound on L z ˜ ; many bounds on M 1 can be found in [7]. Also, from the aforementioned proposition, it is concluded that every lower/upper bound on F provides a(n) lower/upper bound on L z ˜ , respectively; several bounds on F-index can be found in [22].
Proposition 2.
For any graph G with size m and order n, the following identity holds:
L z ˜ ( G ) = 2 ( n 1 ) 2 m L z ( G ) ( n 1 ) M 1 ( G ) .
Proof. 
The identity given in Proposition 1 gives the desired result after utilizing Identity (1). □
By Proposition 2, every upper bound on L z provides a lower bound on L z ˜ and every lower bound on L z gives an upper bound on L z ˜ ; a considerable number of bounds on L z can be found in [22].

3. Extremal Results Concerning ξ -Cyclic Graphs

If s t E ( G ) and r t E ( G ) , then let G s t + r t denote the graph formed from G by removing the edge s t and adding the edge r t . We begin this section by providing the following simple but useful lemma that will be used frequently in the remaining part of this paper:
Lemma 1.
Suppose that G is an n-order graph containing r , s , t , such that r t E ( G ) and s t E ( G ) . If G * = G s t + r t , then
L z ˜ ( G ) L z ˜ ( G * ) = ( n 4 ) + 3 ( n d r d s ) ( d r d s + 1 ) ,
where d r = d r ( G ) and d s = d s ( G ) .
Proof. 
Utilizing the definition of L z ˜ , we obtain
L z ˜ ( G ) L z ˜ ( G * ) = d r ( n d r 1 ) 2 + d s ( n d s 1 ) 2 ( d r + 1 ) ( n d r 2 ) 2 ( d s 1 ) ( n d s ) 2 = ( n 4 ) + 3 ( n d r d s ) ( d r d s + 1 ) .
Lemma 2.
Let G be an n-order graph containing a path r s t such that r t E ( G ) and the edge r s does not lie on any cycle of length 3, where d r ( G ) d s ( G ) and n 5 . If G * = G s t + r t ; then,
L z ˜ ( G ) > L z ˜ ( G * ) .
Proof. 
By Lemma 1, we obtain
L z ˜ ( G ) L z ˜ ( G * ) = ( n 4 ) + 3 ( n d r d s ) ( d r d s + 1 ) ,
where d r = d r ( G ) and d s = d s ( G ) . Since the edge r s does not lie on any cycle of length 3, we have n d r d s 0 ; thus, under the given constraints, Equation (4) gives
L z ˜ ( G ) L z ˜ ( G * ) > 0 .
Now, we provide the first extremal result involving the minimum possible value of L z ˜ for trees.
Theorem 1.
In the set of all n-order trees, with n 5 , only the star graph S n possesses the lowest value of L z ˜ ; the mentioned lowest value is ( n 1 ) ( n 2 ) 2 .
Proof. 
Let T be a tree possessing the lowest value of L z ˜ in the set of all n-order trees. Suppose on the contrary that T S n . Then, Δ ( T ) n 1 . Consider a path r s t of T such that d r ( T ) d s ( T ) . If T * is the graph deduced from T by dropping s t and inserting r t , then by Lemma 2 we have
L z ˜ ( T ) > L z ˜ ( T * ) ,
a contradiction. Also, by elementary computations, one has
L z ˜ ( S n ) = ( n 1 ) ( n 2 ) 2 .
Next, we pay attention to deriving extremal results involving the minimum possible value of L z ˜ for connected ξ -cyclic graphs. For this, we require the next two results.
Lemma 3.
If G is an n-order connected ξ-cyclic graph, r s is an edge of G, and γ is the number of common neighbors of s and r, then d r + d s n + γ n + ξ .
Proof. 
Let α be the number of those neighbors of r that are neither adjacent to s nor equal to s; see Figure 1. Let β be the number of those neighbors of s that are neither adjacent to r nor equal to r. Then, d r = α + γ + 1 and d s = β + γ + 1 . Note that α + β + γ n 2 and γ ξ . Thus,
d r + d s = α + β + 2 γ + 2 n + γ n + ξ .
Lemma 4.
Suppose that G is a connected ξ-cyclic graph of order n containing a path r s t such that r t E ( G ) , where d r ( G ) d s ( G ) and n 3 ξ + 5 . If G * = G s t + r t , then
L z ˜ ( G ) > L z ˜ ( G * ) .
Proof. 
In the following, we take d r = d r ( G ) and d s = d s ( G ) . By Lemma 3, the inequality n d r d s ξ holds, and thus under the given constraints, Lemma 1 yields
L z ˜ ( G ) L z ˜ ( G * ) = ( n 4 ) + 3 ( n d r d s ) ( d r d s + 1 ) > 0 .
Corollary 1.
Let G be a graph possessing the lowest value of L z ˜ in the set of all n-order connected ξ-cyclic graphs, with n 3 ξ + 5 . Then, Δ ( G ) = n 1 .
Proof. 
Contrarily, assume that Δ ( G ) n 1 . Take a vertex r V ( G ) satisfying d r ( G ) = Δ ( G ) . Then, G has a path r s t such that r t E ( G ) . Take G * = G s t + r t . By Lemma 4, it holds that L z ˜ ( G ) > L z ˜ ( G * ) , which is not possible because of the definition of G. Thus, Δ ( G ) = n 1 . □
Next, we provide extremal results involving the minimum possible values of L z ˜ for connected ξ -cyclic graphs when ξ = 1 , 2 , 3 .
Note that there are only two (non-isomorphic) 4-order connected unicyclic graphs, and both of them have the same value of L z ˜ . Thus, in the next theorem, we find the extremal graphs of the order of at least 5.
Theorem 2.
The graph generated from the n-order start graph S n by inserting an edge solely possesses the lowest value of L z ˜ in the set of all n-order connected unicyclic graphs, for every n 8 . For n = 7 , 6 , 5 , the extremal graphs are depicted in the first row, second row, and third row of Figure 2, respectively.
Proof. 
Since there is only one n-order unicyclic graph of maximum degree n 1 for every n 8 , the desired conclusion follows from Corollary 1 for n 8 . Lemma 2 implies that if G is a graph possessing the least value of L z ˜ in the set of all n-order connected unicyclic graphs, with n 5 , then the number of pendent vertices in G becomes n 3 . Simple calculations yield the desired conclusion for n = 5 , 6 , 7 . □
For n 5 , denote by B n ( 1 ) ( η 1 , η 2 , η 3 , η 4 ) the n-order bicyclic graph shown in Figure 3, where η 1 + η 2 + η 3 + η 4 = n 4 with η 1 η 4 0 and η 2 η 3 0 . Also, for n 5 , denote by B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) the n-order bicyclic graph shown in Figure 4, where η 1 + η 2 + η 3 + η 4 + η 5 = n 5 with η 1 max { η 2 , η 3 , η 4 } and η i 0 for every i { 1 , 2 , , 5 } .
Theorem 3.
Considering the family of all n-order bicyclic connected graphs having n 11 , only the graph B n ( 1 ) ( n 4 , 0 , 0 , 0 ) possesses the lowest value of L z ˜ .
Proof. 
Let B n be an n-order connected bicyclic graph, with n 11 . If Δ ( B n ) n 1 , then Corollary 1 implies that B n cannot possess the lowest value of L z ˜ in the set of all n-order connected bicyclic graphs. If Δ ( B n ) = n 1 , then
B n { B n ( 1 ) ( n 4 , 0 , 0 , 0 ) , B n ( 2 ) ( 0 , 0 , 0 , 0 , n 5 ) } .
By elementary calculations, we verify that the inequality
L z ˜ B n ( 1 ) ( n 4 , 0 , 0 , 0 ) < L z ˜ B n ( 2 ) ( 0 , 0 , 0 , 0 , n 5 )
holds for n 11 . Hence, for every n 11 , it holds that
L z ˜ ( B n ) > L z ˜ ( B n ( 1 ) ( n 4 , 0 , 0 , 0 ) ) whenever B n B n ( 1 ) ( n 4 , 0 , 0 , 0 ) .
Next, we characterize the graphs possessing the lowest value of L z ˜ from the set of graphs mentioned in Theorem 3 for 5 n 10 .
Theorem 4.
Let B n be an n-order connected bicyclic graph with n 8 such that the cycles of B n do not share any edge and Δ ( B n ) n 1 . Then, B n does not possess the lowest value of L z ˜ in the set of all n-order connected bicyclic graphs for every n 8 .
Proof. 
Take a vertex r V ( B n ) satisfying d r ( B n ) = Δ ( B n ) . Since Δ ( B n ) n 1 , the graph B n has a path r s t such that r t E ( B n ) . Form a new graph B n * from B n by dropping s t and inserting r t . By Lemma 3, the inequality n d r ( B n ) d s ( B n ) 1 holds, and thus by bearing in mind the given assumptions and Lemma 1, we obtain
L z ˜ ( B n ) L z ˜ ( B n * ) = ( n 4 ) + 3 n d r ( B n ) d s ( B n ) d r ( B n ) d s ( B n ) + 1 > 0 ,
which implies that B n cannot possess the lowest value of L z ˜ in the set of all n-order connected bicyclic graphs for every n 8 . □
Lemma 2 implies the next corollary.
Corollary 2.
If G is a graph possessing the least value of L z ˜ in the set of all n-order connected bicyclic graphs, with n 5 , then G is isomorphic to either B n ( 1 ) ( η 1 , η 2 , η 3 , η 4 ) or B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) (see Figure 3 and Figure 4).
Lemma 5.
For the n-order bicyclic graph B n ( 1 ) ( η 1 , η 2 , η 3 , η 4 ) (depicted in Figure 3) with η 2 η 3 1 and n 6 , the following inequality holds:
L z ˜ ( B n ( 1 ) ( η 1 , η 2 + 1 , η 3 1 , η 4 ) ) < L z ˜ ( B n ( 1 ) ( η 1 , η 2 , η 3 , η 4 ) ) .
Proof. 
Since η 1 + η 2 + η 3 + η 4 = n 4 with η 1 η 4 0 , it holds that η 2 + η 3 n 4 . Therefore, by bearing in mind the given assumptions and Lemma 1, we obtain
L z ˜ ( B n ( 1 ) ( η 1 , η 2 , η 3 , η 4 ) ) L z ˜ ( B n ( 1 ) ( η 1 , η 2 + 1 , η 3 1 , η 4 ) ) = ( n 4 ) + 3 ( n η 2 η 3 4 ) ( η 2 η 3 + 1 ) > 0 .
Lemma 6.
For the n-order bicyclic graph B n ( 1 ) ( η 1 , η 2 , 0 , η 4 ) (depicted in Figure 3) with η 1 η 4 1 and 6 n 10 , it holds that
L z ˜ ( B n ( 1 ) ( η 1 + 1 , η 2 , 0 , η 4 1 ) ) = L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , η 4 ) ) if either η 2 = 1 and n = 7 , or η 2 = 0 and n = 10 , < L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , η 4 ) ) if η 2 1 and 8 n 10 , > L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , η 4 ) ) if η 2 = 0 and 6 n 9 .
Proof. 
By Lemma 1, we obtain
L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , η 4 ) ) L z ˜ ( B n ( 1 ) ( η 1 + 1 , η 2 , 0 , η 4 1 ) ) = ( n 4 ) + 3 ( n η 1 η 4 6 ) ( η 1 η 4 + 1 ) .
If η 2 1 , then the equation η 1 + η 2 + η 4 = n 4 gives η 1 + η 4 n 5 (and hence n 7 ; if n = 7 then η 1 = η 2 = η 4 = 1 ) and thus under the given constraints Equation (5) yields
L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , η 4 ) ) L z ˜ ( B n ( 1 ) ( η 1 + 1 , η 2 , 0 , η 4 1 ) ) = 0 if n = 7 , > 0 if 8 n 10 .
If η 2 = 0 , then η 1 + η 4 = n 4 and thus Equation (5) yields the desired conclusion. □
Lemma 7.
For the n-order bicyclic graph B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) with n 6 , the following relations hold:
L z ˜ ( B n ( 1 ) ( η 1 + 1 , η 2 1 , 0 , 0 ) ) < L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) ) if η 1 η 2 1 0 and n 8 , = L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) ) if either n = 7 and η 2 1 , or η 1 = η 2 2 and η 2 1 , > L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) ) if either η 2 = η 1 = 1 and n = 6 , or η 1 < η 2 2 and n 8 .
and
L z ˜ ( B n ( 1 ) ( η 1 1 , η 2 + 1 , 0 , 0 ) ) < L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) ) if either η 2 1 η 1 1 and n 8 , or η 2 = 0 , η 1 = 2 and n = 6 , = L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) ) if either n = 7 and η 1 1 , or η 2 = η 1 1 , > L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) ) if η 2 < η 1 and n 8 .
Proof. 
Since η 1 + η 2 = n 4 , we have
L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) ) L z ˜ ( B n ( 1 ) ( η 1 + 1 , η 2 1 , 0 , 0 ) ) = ( n 7 ) ( η 1 η 2 + 2 )
and
L z ˜ ( B n ( 1 ) ( η 1 , η 2 , 0 , 0 ) ) L z ˜ ( B n ( 1 ) ( η 1 1 , η 2 + 1 , 0 , 0 ) ) = ( n 7 ) ( η 2 η 1 ) ,
which yield the desired relations. □
Lemma 8.
For the n-order bicyclic graph B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) (depicted in Figure 4) with n 7 , the following inequalities hold:
L z ˜ ( B n ( 2 ) ( η 1 + 1 , η 2 , η 3 , η 4 1 , η 5 ) ) < L z ˜ ( B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) ) when η 4 1 ,
L z ˜ ( B n ( 2 ) ( η 1 + 1 , η 2 , η 3 1 , η 4 , η 5 ) ) < L z ˜ ( B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) ) when η 3 1 ,
L z ˜ ( B n ( 2 ) ( η 1 + 1 , η 2 1 , η 3 , η 4 , η 5 ) ) < L z ˜ ( B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) ) when η 2 1 ,
Proof. 
Since η 1 + η 2 + η 3 + η 4 + η 5 = n 5 with η i 0 for i { 1 , 2 , , 5 } , it holds that η 1 + η j n 5 for every j { 2 , , 5 } . Also, we recall that η 1 max { η 2 , η 3 , η 4 } . Therefore, by using the given constraints, we have
L z ˜ ( B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) ) L z ˜ ( B n ( 2 ) ( η 1 + 1 , η 2 1 , η 3 , η 4 , η 5 ) ) = ( n 4 ) + 3 ( n η 1 η 2 4 ) ( η 1 η 2 + 1 ) > 0 ,
L z ˜ ( B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) ) L z ˜ ( B n ( 2 ) ( η 1 + 1 , η 2 , η 3 1 , η 4 , η 5 ) ) = ( n 4 ) + 3 ( n η 1 η 3 4 ) ( η 1 η 3 + 1 ) > 0 ,
L z ˜ ( B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) ) L z ˜ ( B n ( 2 ) ( η 1 + 1 , η 2 , η 3 , η 4 1 , η 5 ) ) = ( n 4 ) + 3 ( n η 1 η 4 4 ) ( η 1 η 4 + 1 ) > 0 .
Lemma 9.
For the n-order bicyclic graph B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) , the following relations hold:
L z ˜ ( B n ( 2 ) ( η 1 1 , 0 , 0 , 0 , η 5 + 1 ) ) < L z ˜ ( B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) ) if 1 η 1 η 5 + 2 and n 8 , = L z ˜ ( B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) ) if either n = 7 and η 1 1 , or η 5 + 3 = η 1 1 , > L z ˜ ( B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) ) if η 1 = 1 , η 5 = 0 and n = 6 , or η 5 + 3 < η 1 and n 8 .
and
L z ˜ ( B n ( 2 ) ( η 1 + 1 , 0 , 0 , 0 , η 5 1 ) ) < L z ˜ ( B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) ) if η 1 2 η 5 1 and n 8 , or η 1 = 0 , η 5 = 1 and n = 6 , = L z ˜ ( B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) ) if either n = 7 and η 5 1 , or η 1 1 = η 5 1 , > L z ˜ ( B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) ) if η 5 1 , η 1 1 < η 5 and n 8 .
Proof. 
Since η 1 + η 5 = n 5 , we have
L z ˜ ( B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) ) L z ˜ ( B n ( 2 ) ( η 1 1 , 0 , 0 , 0 , η 5 + 1 ) ) = ( n 7 ) ( η 5 η 1 + 3 )
and
L z ˜ ( B n ( 2 ) ( η 1 , 0 , 0 , 0 , η 5 ) ) L z ˜ ( B n ( 2 ) ( η 1 + 1 , 0 , 0 , 0 , η 5 1 ) ) = ( n 7 ) ( η 1 η 5 1 ) ,
from which the required relations follow. □
Denote by K 4 e the graph deduced from the 4-order complete graph K 4 by dropping an edge. Let B n ; α be the n-order graph formed from K 4 e by connecting α pendent vertices with one vertex having degree 3 and the rest n 4 α pendent vertices with the other vertex having degree 3 (see the graph placed on the left-hand side in Figure 5), where 0 α ( n 4 ) / 2 . Let B n * denote the graph generated from K 4 e by attaching n 4 pendent vertices to a vertex of degree 2 (see the graph placed on the right-hand side in Figure 5). Certainly, B n ; 0 = B n ( 1 ) ( n 4 , 0 , 0 , 0 ) .
Now, we are in the position to characterize the graphs possessing the lowest value of L z ˜ from the set of graphs mentioned in Theorem 3 for 5 n 10 .
Theorem 5.
Among all n-order connected bicyclic graphs,
(i) 
only the graph B n ( 1 ) ( 0 , n 4 , 0 , 0 ) has the minimum value of L z ˜ for n = 5 ,
(ii) 
only the graph B n ; 1 has the minimum value of L z ˜ for each n { 6 , 7 } ,
(iii) 
only the graph B n ; 2 has the minimum value of L z ˜ for each n { 8 , 9 } ,
(iv) 
only the graphs B n ; 0 , B n ; 1 , B n ; 2 , B n ; 3 have the minimum value of L z ˜ for n = 10 .
Proof. 
By Corollary 2, it is enough to investigate the values of L z ˜ for the graphs:
B n ( 1 ) ( η 1 , η 2 , η 3 , η 4 ) and B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) .
For n = 5 , there are only three such graphs; namely, B n ( 1 ) ( n 4 , 0 , 0 , 0 ) , B n ( 1 ) ( 0 , n 4 , 0 , 0 ) , and B n ( 2 ) ( 0 , 0 , 0 , 0 , n 5 ) . Certainly, for n = 5 , it holds that
L z ˜ ( B n ( 1 ) ( 0 , n 4 , 0 , 0 ) ) < L z ˜ ( B n ( 1 ) ( n 4 , 0 , 0 , 0 ) ) < L z ˜ ( B n ( 2 ) ( 0 , 0 , 0 , 0 , n 5 ) ) ,
which confirms Part (i) of the theorem.
If G is a graph possessing the least value of L z ˜ in the set of all 6-order connected bicyclic graphs, then from Corollary 2 and Lemmas 5–9, it follows that G { G 1 , G 2 , G 3 , G 4 } where the graphs G 1 , G 2 , G 3 , G 4 are depicted in Figure 6. It holds that
L z ˜ ( G 1 ) < L z ˜ ( G 2 ) = L z ˜ ( G 3 ) < L z ˜ ( G 4 ) .
If G is a graph possessing the least value of L z ˜ in the set of all 7-order connected bicyclic graphs, then from Corollary 2 and Lemmas 5–9, it follows that G { H 1 , H 2 , , H 8 } where the graphs H 1 , H 2 , , H 8 are depicted in Figure 7. It holds that
L z ˜ ( H 1 ) < L z ˜ ( H 2 ) = L z ˜ ( H 3 ) = L z ˜ ( H 5 ) = L z ˜ ( H 6 )
and
L z ˜ ( H 1 ) < L z ˜ ( H 4 ) = L z ˜ ( H 7 ) = L z ˜ ( H 8 ) .
Since G 1 = B 6 ; 1 and H 1 = B 7 ; 1 , the proof of Part (ii) is completed.
Theorem 4 states that if B n is an n-order connected bicyclic graph with n 8 such that the cycles of B n do not share any edge and Δ ( B n ) n 1 ; then, B n does not possess the lowest value of L z ˜ in the family of all n-order bicyclic connected graphs having n 8 . Thus, if G is a graph possessing the least value of L z ˜ in the set of all n-order connected bicyclic graphs for n { 8 , 9 , 10 } , then from Corollary 2 and Lemmas 5–7 it follows that
G { B n ; 0 , B n ; 2 , B n * , B n } when n { 8 , 9 } , G { B n ; α , B n * , B n } when n = 10 .
where B n = B n ( 2 ) ( 0 , 0 , 0 , 0 , n 5 ) . For n { 8 , 9 , 10 } , the following relations hold:
L z ˜ ( B n ; 0 ) = n 3 n 2 28 n + 68 < L z ˜ ( B n * ) = n 3 39 n + 96 < L z ˜ ( B n ) = n 3 n 2 24 n + 52 .
Also, the inequality L z ˜ ( B n ; 2 ) < L z ˜ ( B n ; 0 ) holds for each n { 8 , 9 } . Moreover, the equation L z ˜ ( B 10 ; α ) = 688 holds for every α { 0 , 1 , 2 , 3 } . Thus, Parts (iii) and (iv) also hold. □
Theorem 6.
Considering the family of all n-order tricyclic connected graphs with n 14 , only the right-most graph in Figure 8 possesses the lowest value of L z ˜ .
Proof. 
Let G be an n-order connected tricyclic graph, with n 14 . If Δ ( G ) n 1 , then Corollary 1 implies that G cannot possess the lowest value of L z ˜ in the set of all n-order connected tricyclic graphs. If Δ ( G ) = n 1 , then G is one of the graphs J 1 , J 2 , , J 5 (from left to right, respectively) shown in Figure 8. We have
L z ˜ ( J 1 ) = ( n 7 ) ( n 2 ) 2 + 12 ( n 3 ) 2 , L z ˜ ( J 2 ) = ( n 6 ) ( n 2 ) 2 + 8 ( n 3 ) 2 + 3 ( n 4 ) 2 , L z ˜ ( J 3 ) = ( n 5 ) ( n 2 ) 2 + 4 ( n 3 ) 2 + 6 ( n 4 ) 2 , L z ˜ ( J 4 ) = ( n 5 ) ( n 2 ) 2 + 6 ( n 3 ) 2 + 4 ( n 5 ) 2 , L z ˜ ( J 5 ) = ( n 4 ) ( n 2 ) 2 + 9 ( n 4 ) 2 .
But, L z ˜ ( J i ) > L z ˜ ( J 5 ) for every i { 1 , 2 , 3 , 4 } because n 14 . Hence, L z ˜ ( G ) L z ˜ ( J 5 ) with equality if and only if G = J 5 . □
A non-trivial path P : r 1 r p of a graph G is said to be a pendent path if
max { d r 1 ( G ) , d r p ( G ) } 3 and min { d r 1 ( G ) , d r p ( G ) } = 1 ,
provided that d r i ( G ) = 2 when 2 i p 1 . By adjacent pendent paths in a graph G, we mean the pendent paths of G having a common vertex.
Lemma 10.
For n 5 , if G is a connected n-order graph possessing adjacent pendent paths, then there a connected n-order graph G * exists that has no adjacent pendent paths satisfying | E ( G ) | = | E ( G * ) | and
L z ˜ ( G ) < L z ˜ ( G * ) .
Proof. 
Let s be the common vertex of two adjacent pendent paths P 1 and P 2 in G. Assume that the edge s t belongs to the path P 1 . Let r V ( G ) be the vertex of P 2 satisfying d r ( G ) = 1 . Take G = G s t + r t . Evidently, | E ( G ) | = | E ( G ) | . Since n 4 > 0 and n 1 d s ( G ) 3 , by Lemma 1 we obtain
L z ˜ ( G ) L z ˜ ( G ) = [ n 4 + 3 ( n 1 d s ( G ) ) ] ( d s ( G ) 2 ) < 0 .
If G contains no adjacent pendent paths, the lemma holds true. If G does contain such paths, we can again perform the above-mentioned transformation successively until we obtain the desired graph G * satisfying L z ˜ ( G ) < L z ˜ ( G ) < < L z ˜ ( G * ) . □
The next result is one of the direct outcomes of Lemma 10.
Theorem 7.
In the set of all n-order trees, with n 5 , only the path graph P n possesses the highest value of L z ˜ ; the mentioned highest value is
2 ( n 2 ) ( n 2 5 n + 7 ) .
Lemma 11.
If G is a graph possessing the highest value of L z ˜ among the family of all n-order connected ξ-cyclic graphs admitting n 5 and ξ 1 , then δ ( G ) > 1 .
Proof. 
Contrary, let δ ( G ) = 1 . Because of the constraint ξ 1 , the graph G must contain at least one pendent path, say P. Let r and s be the terminal vertices of the path P; particularly, assume that d s ( G ) 3 and d r ( G ) = 1 . Let s t E ( G ) , where t does not belong to the path P. Take G * = G s t + r t . After the same calculations as made in the proof of Lemma 10, we arrive at L z ˜ ( G ) < L z ˜ ( G * ) ; this contradicts the definition of G. Thereby, δ ( G ) > 1 . □
The next result is one of the direct outcomes of Lemma 11.
Theorem 8.
Considering the family of all n-order unicyclic connected graphs having n 5 , only the cycle graph C n possesses the highest value of L z ˜ ; the mentioned highest value is 2 n ( n 3 ) 2 .
Lemma 12.
If G is a graph possessing the highest value of L z ˜ in the family of all n-order connected ξ-cyclic graphs admitting n 2 ( ξ 1 ) 2 and n 8 , then Δ ( G ) = 3 .
Proof. 
The connectedness of G and the constraint ξ 2 guarantee that Δ ( G ) 3 . Contrarily, let Δ ( G ) 4 . By Lemma 11, the inequality δ ( G ) 2 holds.
Suppose that G has m edges. Represent by N i the number of members of { r V ( G ) : d r ( G ) = i } . Since ξ = m n + 1 , the inequality n 2 ( ξ 1 ) yields n 2 ( m n ) , which further implies that
i = 2 Δ ( G ) N i 2 i = 2 Δ ( G ) i N i 2 i = 2 Δ ( G ) N i = 2 i = 3 Δ ( G ) i N i 2 i = 3 Δ ( G ) N i ,
which guaranties that
N 2 i = 4 Δ ( G ) ( i 3 ) N i ;
that is, G possess at least one vertex with degree 2.
Consider a vertex s V ( G ) such that d s ( G ) = Δ ( G ) (then, we certainly have 4 d s ( G ) n 1 ). Also consider a vertex r V ( G ) having degree 2. The inequality d s ( G ) 4 confirms the existence of no less than two neighbors of s that are not adjacent to r. We pick from these neighbors of s a vertex t such that the graph G * = G s t + r t is connected. First, by using Lemma 1 and then by using the inequalities 4 d s ( G ) n 1 and n 8 , we obtain
L z ˜ ( G ) L z ˜ ( G * ) = [ n 7 + 3 ( n 1 d s ( G ) ) ] ( d s ( G ) 3 ) < 0 ,
which is at odds with the definition of G. Therefore, we derive that Δ ( G ) = 3 , as desired. □
Theorem 9.
Consider the set G n , ξ of all n-order connected ξ-cyclic graphs with n 2 ( ξ 1 ) 2 and n 8 .
(i). 
If n = 2 ( ξ 1 ) , then only (the) 3-regular graph(s) possess(es) the highest value of L z ˜ in G n , ξ .
(ii). 
If n > 2 ( ξ 1 ) , then only the graphs with ( Δ , δ ) = ( 3 , 2 ) possess the highest value of L z ˜ in G n , ξ .
Proof. 
Assume that G is a graph possessing the highest value of L z ˜ in G n , ξ . Then, by Lemmas 11 and 12, it holds that δ ( G ) 2 and Δ ( G ) = 3 . Thus, we have
N 2 + N 3 = n
and
2 N 2 + 3 N 3 = 2 ( n + ξ 1 ) ,
where N i is defined in the proof of Lemma 12.
(i)
If n = 2 ( ξ 1 ) , then Equations (6) and (7) yield N 2 = 0 and thus G is 3-regular.
(ii)
If n > 2 ( ξ 1 ) , then Equations (6) and (7) imply that N 2 > 0 and N 3 > 0 , as desired.
One of the implications of Theorem 9 is the following result.
Theorem 10.
Only the graphs with ( Δ , δ ) = ( 3 , 2 ) possess the highest value of L z ˜ in the family of all n-order bicyclic connected graphs admitting n 6 . For n = 5 , such an extremal graph can be constructed from the star graph through inserting two non-adjacent edges.
Proof. 
If n 8 , then the desired conclusion follows from Theorem 9. If n 5 , then by Lemma 11, the minimum degree of a graph possessing the highest value of L z ˜ in the set of all n-order connected bicyclic graphs must be at least 2. If n = 5 , then there are only three n-order connected bicyclic graphs with minimum degree of at least 2; Figure 9 shows all these three graphs together with the values of L z ˜ . Now, in the following, assume that n { 6 , 7 } .
Assume that G is a graph possessing the highest value of L z ˜ in the family of all n-order connected bicyclic graphs. Then, by Lemma 11, it holds that δ ( G ) 2 . We claim that Δ ( G ) = 3 . We note that there is no bicyclic graph with Δ = n 1 and δ 2 (because n 6 ). Thus, from the proof of Lemma 12 it follows that Δ ( G ) = 3 . Since n > 2 ( ξ 1 ) = 2 , from the proof of Theorem 9(ii) it follows that δ ( G ) = 2 . □
Another implication of Theorem 9 is the following theorem.
Theorem 11.
Only the graphs with ( Δ , δ ) = ( 3 , 2 ) possess the highest value of L z ˜ in the family of all n-order tricyclic connected graphs admitting n 7 . For n = 5 , such an extremal graph can be constructed from the star graph through inserting three edges between one fixed pendent vertex and three other pendent vertices (see the right-most graph in Figure 10). For n = 6 , the set of such extremal graphs consists of the graphs for which ( Δ , δ ) { ( 3 , 2 ) , ( 5 , 2 ) } .
Proof. 
If n 8 , then the desired conclusion follows from Theorem 9. If n 5 , then by Lemma 11 the minimum degree of a graph possessing the highest value of L z ˜ in the set of all n-order connected tricyclic graphs must be at least 2. If n = 5 , then there are only three n-order connected tricyclic graphs with a minimum degree at least 2; Figure 10 shows all these three graphs, and its caption gives the values of L z ˜ for the mentioned three graphs. Now, in the following, assume that n { 6 , 7 } .
Assume that G is a graph possessing the highest value of L z ˜ in the family of all n-order tricyclic connected graphs. Then, by Lemma 11, it holds that δ ( G ) 2 . First, let Δ ( G ) < n 1 . Then, from the proof of Lemma 12, it follows that Δ ( G ) = 3 . Since n > 2 ( ξ 1 ) = 4 , from the proof of Theorem 9(ii) it follows that δ ( G ) = 2 . Equations (6) and (7) yield N 2 = n 4 and N 3 = 4 . Thus, we have
L z ˜ ( G ) = 2 ( n 4 ) ( n 2 15 ) .
On the other hand, if Δ ( G ) = n 1 then G G , where G is the n-order connected tricyclic graph with maximum degree n 1 and minimum degree at least 2 (see Figure 11). Here,
L z ˜ ( G ) = 2 ( n 4 ) ( n 2 15 ) if n = 6 , < 2 ( n 4 ) ( n 2 15 ) if n = 7 .
This completes the proof. □

4. Extremal Results Concerning Molecular ξ -Cyclic Graphs

Note that the extremal graphs specified in Theorems 9–11 are molecular ones, except for n = 6 in Theorem 11. Thus, these graphs remain extremal if one puts the following additional constraint on the graphs considered in these theorems: the maximum degree is at most 4. Also, from Theorem 11 we deduce that the graphs with ( Δ , δ ) = ( 3 , 2 ) are the only graphs possessing the highest ad-hoc Lanzhou index in the set of all n-order molecular connected tricyclic graphs for n = 6 . Next, we turn our attention to the results concerning the minimum ad-hoc Lanzhou index of molecular ( n , m ) -graphs (or, equivalently n-order ξ -cyclic graphs). An ( n , m ) -graph is an n-order graph of size m. For a graph G, define
H f ( G ) = s V ( G ) f ( d s ) .
where f is a real-valued function. Some initial studies, recent developments, and a survey on the indices of the form (8) can be found in [23,24,25,26,27,28,29], respectively.
Lemma 13
([30]). Consider a molecular ( n , m ) -graph G, where n 5 . Take
ψ 1 = 2 3 f ( 1 ) + f ( 2 ) 1 3 f ( 4 ) and ψ 2 = 1 3 f ( 1 ) + f ( 3 ) 2 3 f ( 4 ) .
If min { ψ 1 , ψ 2 } > 0 and ψ 2 / 2 < ψ 1 < 2 ψ 2 , then
H f ( G ) 1 3 4 f ( 1 ) f ( 4 ) n + 2 3 f ( 4 ) f ( 1 ) m + f ( 2 ) 2 3 f ( 1 ) 1 3 f ( 4 ) if 2 m n 1 ( mod 3 ) f ( 3 ) 1 3 f ( 1 ) 2 3 f ( 4 ) if 2 m n 2 ( mod 3 ) 0 if 2 m n 0 ( mod 3 )
where the equality holds if and only if the degree set of G is
  • { 1 , 2 , 4 } and G admits exactly one vertex of degree 2 whenever 2 m n 1 ( mod 3 ) ;
  • { 1 , 3 , 4 } and G admits exactly one vertex of degree 3 whenever 2 m n 2 ( mod 3 ) ;
  • { 1 , 4 } whenever 2 m n 0 ( mod 3 ) .
Theorem 12.
For a molecular ( n , m ) -graph G, with n 8 , the following holds:
L z ˜ ( G ) 2 m ( n 8 ) ( n 4 ) + 4 n ( 2 n 7 ) + 2 ( 2 n 9 ) if 2 m n 1 ( mod 3 ) 4 ( n 5 ) if 2 m n 2 ( mod 3 ) 0 if 2 m n 0 ( mod 3 )
where the equality characterization is the same as mentioned in Lemma 13.
Proof. 
Consider f ( x ) = x ( n 1 x ) 2 with n 8 . Then, we have
2 3 f ( 1 ) + f ( 2 ) 1 3 f ( 4 ) > 0 ,
1 3 f ( 1 ) + f ( 3 ) 2 3 f ( 4 ) > 0 ,
and
1 2 1 3 f ( 1 ) + f ( 3 ) 2 3 f ( 4 ) < 2 3 f ( 1 ) + f ( 2 ) 1 3 f ( 4 ) < 2 1 3 f ( 1 ) + f ( 3 ) 2 3 f ( 4 ) .
Thus, by Lemma 13, the desired result follows. □
One of the simple but noticeable consequences of Theorem 12 is the following extremal result involving molecular trees.
Corollary 3.
For every n 8 , in the family of all n-order molecular trees, only the trees having the degree set
(i) 
{ 1 , 2 , 4 } and admitting exactly one vertex of degree 2 possess the least value of L z ˜ , when n 0 ( mod 3 ) ;
(ii) 
{ 1 , 3 , 4 } and admitting exactly one vertex of degree 3 possess the least value of L z ˜ , when n 1 ( mod 3 ) ;
(iii) 
{ 1 , 4 } possess the least value of L z ˜ , when n 2 ( mod 3 ) .

5. Extremal Results for n -Order Graphs

To prove the first extremal result involving L z ˜ for n-order graphs, we require the following known result:
Lemma 14
([10]). If G is an n-order graph, then
0 L z ( G ) 4 n ( n 1 ) 3 27 .
Here, the left equality is true if and only if G { K n , K ¯ n } , and the right equality is true if and only if n 1 ( mod 3 ) and G is a 2 ( n 1 ) 3 - regular graph.
Proposition 3.
In the set of all n-order graphs, only the edgeless graph K ¯ n and the complete graph K n possess the least value of L z ˜ ; the mentioned least value is 0. Also, in the same set with the constraint n 1 ( mod 3 ) , only the ( n 1 ) 3 -regular graph possesses the highest value of L z ˜ ; the mentioned highest value is
4 n ( n 1 ) 3 27 .
Proof. 
Since L z ( G ¯ ) = L z ˜ ( G ) , by Lemma 14 it is enough to show the existence of at least one ( n 1 ) 3 -regular graph with n vertices satisfying the congruence n 1 ( mod 3 ) . Since n 1 ( mod 3 ) , we have n 1 = 3 k for some integer k. Thus, if n 1 is even then k must be even, and thereby n 1 3 remains even. Eventually, whether n 1 is even or odd, in either case, we conclude that n ( n 1 ) 3 is even. Also, it is a well-known observation that at least one t-regular graph with order n exists whenever t n is even; this fact implies that at least one ( n 1 ) 3 -regular graph with order n exists for every n satisfying n 1 ( mod 3 ) . □
Next, we pay our attention to the extremum values of L z ˜ for n-order molecular graphs.
Theorem 13.
In the set of all n-order molecular graphs, with the constraint n 14 , only 4-regular graphs possess the highest value of L z ˜ .
Proof. 
Consider an n-order molecular graph G. For every s V ( G ) and n 14 , it holds that d s ( n 1 d s ) 2 4 ( n 5 ) 2 with equality if and only if d s = 4 . Consequently, we have
L z ˜ ( G ) 4 n ( n 5 ) 2 ,
where the equality is true if and only if G is 4-regular. Since at least one 4-regular graph exists for every n 14 , the desired result follows from (9). □
For the minimal version of Theorem 13, we require the following:
Lemma 15.
If G is an n-order molecular graph, with n 14 , and s t E ( G ) , then
L z ˜ ( G ) > L z ˜ ( G s t ) .
Proof. 
Since the function f ( x ) = x ( n 1 x ) 2 , with n 14 , is strictly increasing for 0 x 4 , we obtain
L z ˜ ( G ) L z ˜ ( G s t ) = d s ( n 1 d s ) 2 + d t ( n 1 d t ) 2 ( d s 1 ) ( n 2 d s ) 2 + ( d t 1 ) ( n 2 d t ) 2 > 0 ,
where d s = d s ( G ) and d t = d t ( G ) . □
Theorem 14.
For every n 14 , in the set of all n-order connected molecular graphs, only the trees having the degree set
(i) 
{ 1 , 2 , 4 } and admitting exactly one vertex of degree 2 possess the least value of L z ˜ , when n 0 ( mod 3 ) ;
(ii) 
{ 1 , 3 , 4 } and admitting exactly one vertex of degree 3 possess the least value of L z ˜ , when n 1 ( mod 3 ) ;
(iii) 
{ 1 , 4 } possess the least value of L z ˜ , when n 2 ( mod 3 ) .
Proof. 
Consider an n-order connected molecular graph G containing at least one cycle, where n 14 . By Lemma 15, it holds that L z ˜ ( G ) > L z ˜ ( G s t ) , where s t is an edge lying on a cycle of G. Thus, for every n 14 , a graph possessing the least value of L z ˜ in the set of all n-order connected molecular graphs must be a tree. Consequently, the desired conclusion follows from Corollary 3. □

Author Contributions

Conceptualization, A.A., Y.S., D.D. and T.R.; methodology, A.A., Y.S., D.D. and T.R.; software, A.A., Y.S., D.D. and T.R.; validation, A.A., Y.S., D.D. and T.R.; formal analysis, A.A., Y.S., D.D. and T.R.; investigation, A.A., Y.S., D.D. and T.R.; resources, A.A., Y.S., D.D. and T.R.; data curation, A.A., Y.S., D.D. and T.R.; writing—original draft preparation, A.A., Y.S., D.D. and T.R.; writing—review and editing, A.A., Y.S., D.D. and T.R.; visualization, A.A., Y.S., D.D. and T.R.; supervision, Y.S., D.D. and T.R.; and project administration, Y.S., D.D. and T.R. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The authors can be contacted for details regarding this study’s data.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Gross, J.L.; Yellen, J. Graph Theory and Its Applications, 2nd ed.; CRC Press: Boca Raton, FL, USA, 2005. [Google Scholar]
  2. Wagner, S.; Wang, H. Introduction to Chemical Graph Theory; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
  3. Bondy, J.A.; Murty, U.S.R. Graph Theory; Springer: New York, NY, USA, 2008. [Google Scholar]
  4. Trinajstić, N. Chemical Graph Theory; CRC Press: Boca Raton, FL, USA, 1992. [Google Scholar]
  5. Chartrand, G.; Lesniak, L.; Zhang, P. Graphs & Digraphs, 6th ed.; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
  6. Gutman, I.; Trinajstić, N. Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 1972, 17, 535–538. [Google Scholar] [CrossRef]
  7. Borovićanin, B.; Das, K.C.; Furtula, B.; Gutman, I. Bounds for Zagreb indices. MATCH Commun. Math. Comput. Chem. 2017, 78, 17–100. [Google Scholar]
  8. Furtula, B.; Gutman, I. A forgotten topological index. J. Math. Chem. 2015, 53, 1184–1190. [Google Scholar] [CrossRef]
  9. Su, G.; Wang, S.; Du, J.; Gao, M.; Das, K.C.; Shang, Y. Sufficient conditions for a graph to be -connected, -deficient, -Hamiltonian and --independent in terms of the forgotten topological index. Mathematics 2022, 10, 1802. [Google Scholar] [CrossRef]
  10. Vukičević, D.; Li, Q.; Sedlar, J.; Došlić, T. Lanzhou index. MATCH Commun. Math. Comput. Chem. 2018, 80, 863–876. [Google Scholar]
  11. Alrowaili, D.A.; Zafar, F.; Javaid, M. Characterization of Extremal Unicyclic Graphs with Fixed Leaves Using the Lanzhou Index. Symmetry 2022, 14, 2408. [Google Scholar] [CrossRef]
  12. Li, Q.; Deng, H.; Tang, Z. Lanzhou index of trees and unicyclic graphs. Electron. J. Math. 2023, 5, 29–45. [Google Scholar]
  13. Liu, Q.; Li, Q.; Zhang, H. Unicyclic graphs with extremal Lanzhou index. Appl. Math. J. Chin. Univ. 2022, 37, 350–365. [Google Scholar] [CrossRef]
  14. Saha, L. Lanzhou index of trees with fixed maximum and second maximum degree. MATCH Commun. Math. Comput. Chem. 2022, 88, 593–603. [Google Scholar] [CrossRef]
  15. Dehgardi, N.; Liu, J.B. Lanzhou index of trees with fixed maximum degree. MATCH Commun. Math. Comput. Chem. 2021, 86, 3–10. [Google Scholar]
  16. Ali, A.; Iqbal, Z.; Iqbal, Z. Two physicochemical properties of benzenoid chains: Solvent accessible molecular volume and molar refraction. Can. J. Phys. 2019, 97, 524–528. [Google Scholar] [CrossRef]
  17. Estrada, E. Edge adjacency relationships and a novel topological index related to molecular volume. J. Chem. Inf. Comput. Sci. 1995, 35, 31–33. [Google Scholar] [CrossRef]
  18. Wang, X.; Mao, Y.; Wang, Z.; Wang, S.; Li, J. On the Lanzhou index. MATCH Commun. Math. Comput. Chem. 2021, 85, 109–119. [Google Scholar]
  19. Azari, M.; Falahati-Nezhad, F. Some results on forgotten topological coindex. Iranian J. Math. Chem. 2019, 10, 307–318. [Google Scholar]
  20. Furtula, B.; Gutman, I.; Kovijanić Vukićević, Z.; Lekishvili, G.; Popivoda, G. On an old/new degree-based topological index. Bull. Acad. Serbe Sci. Arts 2015, 40, 19–31. [Google Scholar]
  21. Milovanović, I.; Matejić, M.; Milovanović, E. A note on the general zeroth-order Randić coindex of graphs. Contrib. Math. 2020, 1, 17–21. [Google Scholar]
  22. Liu, J.B.; Matejić, M.M.; Milovanović, E.I.; Milovanović, I.Ž. Some new inequalities for the forgotten topological index and coindex of graphs. MATCH Commun. Math. Comput. Chem. 2020, 84, 719–738. [Google Scholar]
  23. Linial, N.; Rozenman, E. An extremal problem on degree sequences of graphs. Graphs Combin. 2002, 18, 573–582. [Google Scholar] [CrossRef]
  24. Yao, Y.; Liu, M.; Belardo, F.; Yang, C. Unified extremal results of topological index and graph spectrum. Discrete Appl. Math. 2019, 271, 218–232. [Google Scholar] [CrossRef]
  25. Hu, Z.; Li, L.; Li, X.; Peng, D. Extremal graphs for topological index defined by a degree-based edge-weight function. MATCH Commun. Math. Comput. Chem. 2022, 88, 505–520. [Google Scholar] [CrossRef]
  26. Tomescu, I. Graphs with given cyclomatic number extremal relatively to vertex degree function index for convex functions. MATCH Commun. Math. Comput. Chem. 2022, 87, 109–114. [Google Scholar] [CrossRef]
  27. Tomescu, I. Extremal vertex-degree function index for trees and unicyclic graphs with given independence number. Discrete Appl. Math. 2022, 306, 83–88. [Google Scholar] [CrossRef]
  28. Rizwan, M.; Bhatti, A.A.; Javaid, M.; Shang, Y. Conjugated tricyclic graphs with maximum variable sum exdeg index. Heliyon 2023, 9, e15706. [Google Scholar] [CrossRef]
  29. Li, X.; Peng, D. Extremal problems for graphical function-indices and f-weighted adjacency matrix. Discrete Math. Lett. 2022, 9, 57–66. [Google Scholar]
  30. Albalahi, A.M.; Milovanović, I.Ž.; Raza, Z.; Ali, A.; Hamza, A.E. On the vertex-degree-function indices of connected (n,m)-graphs of maximum degree at most four. arXiv 2022, arXiv:2207.00353. [Google Scholar]
Figure 1. The structure of the graph G used in Lemma 3.
Figure 1. The structure of the graph G used in Lemma 3.
Mathematics 11 04256 g001
Figure 2. The graphs possessing the lowest value of L z ˜ in the set of all n-order connected unicyclic graphs, for n = 5 , 6 , 7 .
Figure 2. The graphs possessing the lowest value of L z ˜ in the set of all n-order connected unicyclic graphs, for n = 5 , 6 , 7 .
Mathematics 11 04256 g002
Figure 3. The n-order bicyclic graph B n ( 1 ) ( η 1 , η 2 , η 3 , η 4 ) , where η 1 + η 2 + η 3 + η 4 = n 4 1 with η 1 η 4 0 and η 2 η 3 0 .
Figure 3. The n-order bicyclic graph B n ( 1 ) ( η 1 , η 2 , η 3 , η 4 ) , where η 1 + η 2 + η 3 + η 4 = n 4 1 with η 1 η 4 0 and η 2 η 3 0 .
Mathematics 11 04256 g003
Figure 4. The n-order bicyclic graph B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) , where η 1 + η 2 + η 3 + η 4 + η 5 = n 5 0 with η 1 max { η 2 , η 3 , η 4 } and η i 0 for every i { 1 , 2 , , 5 } .
Figure 4. The n-order bicyclic graph B n ( 2 ) ( η 1 , η 2 , η 3 , η 4 , η 5 ) , where η 1 + η 2 + η 3 + η 4 + η 5 = n 5 0 with η 1 max { η 2 , η 3 , η 4 } and η i 0 for every i { 1 , 2 , , 5 } .
Mathematics 11 04256 g004
Figure 5. The n-order bicyclic graphs B n ; α (on the left side) and B n * (on the right side), where 0 α ( n 4 ) / 2 .
Figure 5. The n-order bicyclic graphs B n ; α (on the left side) and B n * (on the right side), where 0 α ( n 4 ) / 2 .
Mathematics 11 04256 g005
Figure 6. The 6-order connected bicyclic graphs G 1 , G 2 , G 3 , and G 4 .
Figure 6. The 6-order connected bicyclic graphs G 1 , G 2 , G 3 , and G 4 .
Mathematics 11 04256 g006
Figure 7. The 7-order connected bicyclic graphs H 1 , H 2 , , H 8 .
Figure 7. The 7-order connected bicyclic graphs H 1 , H 2 , , H 8 .
Mathematics 11 04256 g007
Figure 8. The graphs J 1 , J 2 , , J 5 , (from left to right, respectively) used in Theorem 6.
Figure 8. The graphs J 1 , J 2 , , J 5 , (from left to right, respectively) used in Theorem 6.
Mathematics 11 04256 g008
Figure 9. All connected bicyclic graphs of order 5 and minimum degree at least 2, together with the values of L z ˜ .
Figure 9. All connected bicyclic graphs of order 5 and minimum degree at least 2, together with the values of L z ˜ .
Mathematics 11 04256 g009
Figure 10. All connected tricyclic graphs of order 5 and minimum degree at least 2. The first, second, and third graphs (from left to right) have the following values of L z ˜ , respectively: 20, 22, and 24.
Figure 10. All connected tricyclic graphs of order 5 and minimum degree at least 2. The first, second, and third graphs (from left to right) have the following values of L z ˜ , respectively: 20, 22, and 24.
Mathematics 11 04256 g010
Figure 11. The n-order connected tricyclic graphs G of maximum degree n 1 and minimum degree no less than 2, for n = 6 , 7 .
Figure 11. The n-order connected tricyclic graphs G of maximum degree n 1 and minimum degree no less than 2, for n = 6 , 7 .
Mathematics 11 04256 g011
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Share and Cite

MDPI and ACS Style

Ali, A.; Shang, Y.; Dimitrov, D.; Réti, T. Ad-Hoc Lanzhou Index. Mathematics 2023, 11, 4256. https://doi.org/10.3390/math11204256

AMA Style

Ali A, Shang Y, Dimitrov D, Réti T. Ad-Hoc Lanzhou Index. Mathematics. 2023; 11(20):4256. https://doi.org/10.3390/math11204256

Chicago/Turabian Style

Ali, Akbar, Yilun Shang, Darko Dimitrov, and Tamás Réti. 2023. "Ad-Hoc Lanzhou Index" Mathematics 11, no. 20: 4256. https://doi.org/10.3390/math11204256

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop