# On-Bond Incident Degree Indices of Square-Hexagonal Chains

Authors: Tariq A. Alraqad; Hicham Saber; Akbar Ali; Jaya Percival Mazorodze
Journal: Journal of Mathematics
 (2022)

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## Abstract

For a graphG, its bond incident degree (BID) index is defined as the sum of the contributions fdu,dv over all edges uv of G, where dw denotes the degree of a vertex w of G and f is a real-valued symmetric function. If fdu,dv=du+dv or dudv, then the corresponding BID index is known as the first Zagreb index M1 or the second Zagreb index M2, respectively. The class of square-hexagonal chains is a subclass of the class of molecular graphs of minimum degree 2. (Formal definition of a square-hexagonal chain is given in the Introduction section). The present study is motivated from the paper (C. Xiao, H. Chen, Discrete Math. 339 (2016) 506–510) concerning square-hexagonal chains. In the present paper, a general expression for calculating any BID index of square-hexagonal chains is derived. The chains attaining the maximum or minimum values of M1 and M2 are also characterized from the class of all square-hexagonal chains having a fixed number of polygons.


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## Body

## 1. Introduction

Those (chemical) graph-theoretical terminologies and notations adopted in the current paper that are not defined here in this paper can be found in some standard (chemical) graph-theoretical books; for example, [1–3]. All the graphs to be considered in the current paper are finite and connected.In what follows, it is assumed thatG is a graph. The edge set and vertex set of G are denoted by EG and VG, respectively. For a vertex w∈VG, its degree is denoted by dwG (or simply by dw whenever there is only one graph under consideration).In chemical graph theory, those graph invariants that have some chemical applicability are often referred to as topological indices. The first and second Zagreb indices [4], appeared in the first half of 1970s (see for example [4, 5]), belong to the most-studied topological indices (especially in chemical graph theory); they are usually denoted by M1 and M2, respectively, and for G, they are defined as follows:(1)M1G=∑u∈VGdu2,M2G=∑uv∈EGdudv.It is known that∑u∈VGdu2=∑uv∈EGdu+dv. Most of their known properties can be found in the review paper [4] and in the related references included therein.ForG, its bond incident degree (BID) index is defined as follows:(2)BIDG=∑uv∈EGfdu,dv=∑1≤a≤b≤ΔGma,bGθa,b,where is the degree of the vertex u, f is a real-valued function such that fdu,dv=fdv,du, uv is the edge du with end vertices u and v of G, ΔG is the maximum degree in G, θa,b=fa,b and ma,bG is the number of those edges of G whose one end vertex has the degree a and the other end vertex has the degree b. We note here that if θa,b=a+b or θa,b=ab, then the corresponding BID index is M1 or M2, respectively. Details about some mathematical aspects of the BID indices can be found in the papers [6–8] as well as in the related references listed therein.A square-hexagonal system is a connected geometric figure formed by concatenating congruent regular squares and/or hexagons side to side in a plane in such a way that the figure divides the plane into one infinite (external) region and a number of finite (internal) regions, and all internal regions must be congruent regular squares and/or hexagons. In a square-hexagonal system, two polygons having a common side are known as adjacent polygons. By inner dual of a square-hexagonal system, we mean a graphID whose vertices are the polygons of the considered square-hexagonal system, while there is an edge between two vertices of ID if and only if the corresponding polygons share a side. A square-hexagonal system is said to be a square-hexagonal chain if its inner dual is the path graph. It should be noted that different square-hexagonal chains may be obtained depending on the polygons’ type and depending on the way how polygons are concatenated. Rnrefers to a square-hexagonal chain consisting of n polygons. If all polygons in Rn are hexagons, then we say that Rn is a hexagonal chain (see, for instance, [9]) and if all the polygons are squares, then Rn is known as a polyomino chain (see, for instance, [10]). Also, if squares and hexagons are concatenated alternately in Rn, then we say that Rn is a phenylene chain (see [11]).Every square-hexagonal chain can be considered as a graph in which the edges represent the sides of the polygons and the vertices represent the points where two sides of a polygon intersect. In the rest of the present paper, by the terminology “square-hexagonal chain(s)” we mean the graph(s) corresponding to the considered square-hexagonal chain(s).Analogous to the definition of square-hexagonal chains, one may give a definition of triangular/square/pentagonal chains. BID indices of triangular/square/pentagonal chains were studied in [10, 12]. The present study can be considered as a continuation of the research conducted in [10, 12] and it is motivated from the paper [13–15] concerning square-hexagonal chains. In the current paper, a general expression for calculating any BID index of square-hexagonal chains is derived. The chains attaining the maximum or minimum values of M1 and M2 are also characterized from the class of all square-hexagonal chains having a fixed number of polygons.


## 2. Main Results

In order to obtain the main results, we require some terminology concerning square-hexagonal chains. In a square-hexagonal chain, a polygon adjacent with only one (two, respectively) other polygon is known as a terminal (nonterminal, respectively) polygon. A nonterminal polygon in the chain is called a kink if its center is not collinear with centers of the two adjacent polygons. In other words, a nonterminal hexagon is a kink if and only if it contains two adjacent vertices of degree two (Figure1) and a nonterminal square is a kink if and only if it contains a vertex of degree two (Figure 2). Following [15], we will consider square-hexagonal chains that contain the following types of kinks:(1)
Kinks of typeT1: A nonterminal hexagon having exactly two adjacent vertices of degree two (see Figure 1);(2)
Kinks of typeT2,1: A nonterminal square containing a vertex of degree two and adjacent to two squares (see Figure 2(a));(3)
Kinks of typeT2,2: A nonterminal square containing a vertex of degree two and adjacent to a square and a hexagon (see Figure 2(b));(4)
Kinks of typeT2,3: A nonterminal square containing a vertex of degree two and adjacent to two hexagons (see Figure 2(c));Figure 1
Kinks of typeT1.Figure 2
(a) Kinks of typeT2,1; (b) Kinks of type T2,2; (c) Kinks of type T2,3.
(a)(b)(c)A square-hexagonal chain is called linear if it has no kinks and it is called a zigzag chain if every nonterminal polygon is a kink. A segment is a maximal linear chain in a square-hexagonal chain, including kinks and/or terminal polygons at its ends.The lengthlS of a segment S is its number of polygons. ES refers to the set of all edges of a segment S. A segment that contains a terminal polygon is known as an external segment. A segment that contains only nonterminal polygons is known as an internal segment. Clearly, a square-hexagonal chain consists of s segments if and only if it contains exactly s−1 kinks.For a square-hexagonalRn, we define the value δRn to be the number of terminal hexagons in Rn. We also define the following values for segments S1,…,Ss of Rn: see Figures 3 and 4.(3)μSi=1,ifSiis an internal segment consisting of a hexagon and a square and containsonly one edge connecting vertices of degree3,0,otherwise.νSi=1,ifSiis an internal segment of length three and contains an edgeconnecting vertices of degree3,0,otherwise.Figure 3
Internal segment consisting of a hexagon and a square and contains an edge connecting vertices of degree 3.Figure 4
Internal segment of length three containing an edge connecting vertices of degree 3.Moreover,μS1=μSs=νS1=νSs=0.(4)τSi=1,ifSiconsists of two squares,0,otherwise.Now, we are ready to establish the general expression for calculating the BID indices of square-hexagonal chains.Theorem 1.
LetRn be square-hexagonal chain containing d squares and n−d hexagons. Suppose that Rn consists of s segments S1,…,Ss and contains α1 kinks of type T1 and α2,j kinks of type T2,j, for j=1,2,3. Then,(5)m2,2Rn=2δRn+α1+2,m2,3Rn=4n−4d+4+2α2,1+α2,2−4δRn−2α1−τS1−τSs−∑i=2s−12τSi−μSi,m3,3Rn=n+2d−5+2δRn+α1−6α2,1−5α2,2−4α2,3+τS1+τSs+∑i=2s−13τSi−μSi+νSi,m2,4Rn=α2,2+2α2,3+τS1+τSs+∑i=2s−12τSi−μSi,m3,4Rn=4α2,1+3α2,2+2α2,3−τS1−τSs−∑i=2s−14τSi−μSi+2νSi,m4,4Rn=∑i=2s−1τSi+νSi.Proof.
LetC1,…,Cs−1 be the kinks of Rn such that Ci joins segments Si and Si+1.
LetA1=EC1, and Ai=ECi\ECi−1, for i=2,…,s−1. Also, we set B1=ES1\EC1, Bs=ESs\ECs−1, and Bj=ESj\ECj−1∪ECj, for j=2,…,s−1.
Clearly, the collectionA1,…,As−1,B1,…,Bs forms a partition for ERn. For 2≤a≤b≤4, i=1,…,s−1, j=1,…,s, let pa,bi be the number of edges in Ai that connects vertices of degrees a and b and qa,bj be the number of edges in Bj that connect vertices of degrees a and b. Then, we get ma,bRn=∑i=1s−1pa,bi+∑j=1sqa,bj.
First, we calculatem2,2Rn. We have q2,21+q2,2s=2δRn+1, and for j=2,…,s−1, we have q2,2j=0. Also, a kink contains an edge joining vertices of degree 2 if and only if it is of type T1. Hence, m2,2=2δRn+α1+2.
To calculatem4,4Rn, note that a segment Si contains an edge joining vertex of degree 4 if and only if Si is an internal segment and τSi=1 or νSi=1. Thus, m4,4Rn=∑i=2s−1τSi+νSi.
Next, we calculatem2,4Rn. Vertices of degree 4 appear only in kinks of type T2,1, T2,2, and T2,3. In fact, p2,41=τS2, p2,4s−1=τSs−1, and for 2≤i≤s−2, p2,4i=τSi+τSi+1 if Ci is a kink of type T2,1, T2,2, or T2,3, and p2,4i=0 otherwise. Hence, ∑i=1s−1pa,bi=2∑i=2s−1τSi. Also, ∑i=1sqa,bi=α2,2+2α2,3+τS1+τSs−∑i=2s−1μSi. Thus,(6)m2,4Rn=α2,2+2α2,3+τS1+τSs+∑i=2s−12τSi−μSi.
Now, each kink of typesT2,1, T2,2, or T2,3 contains exactly one vertex of degree 4, and so the number of vertices of degree 4 is α2,1+α2,2+α2,3. Therefore,(7)m2,4Rn+m3,4Rn+2m4,4Rn=4α2,1+α2,2+α2,3.
Substituting the values ofm2,4Rn and m4,4Rn in (7) and solving the resulting equation for m3,4Rn yield the following:(8)m3,4Rn=4α2,1+α2,2+α2,3−α2,2−2α2,3−τS1−τSs−∑i=2s−12τSi−μSi−2∑i=2s−1τSi+νSi=4α2,1+3α2,2+2α2,3−τS1−τSs−∑i=2s−14τSi−μSi+2νSi.
Similarly, every nonterminal hexagon contains exactly two vertices of degree 2, and a nonterminal square contains a vertex of degree 2 if and only if it is a kink of typeT2,1, T2,2, or T2,3. Now, adding number of vertices of degree 2 in the terminal polygons, we see that the number of vertices of degree 2 in Rn is 2n−2d+4+α2,1+α2,2+α2,3. Hence, 2m2,2Rn+m2,3Rn+m2,4Rn=22n−2d+4+α2,1+α2,2+α2,3. Therefore,(9)m2,3Rn=22n−2d+4+α2,1+α2,2+α2,3−22δRn+α1+2−α2,2−2α2,3−τS1−τSs−∑i=2s−12τSi−μSi=4n−4d+4+2α2,1+α2,2−4δRn−2α1−τS1−τSs−∑i=2s−12τSi−μSi.
The total number of edges inRn is n+1+2d+4n−d=5n−2d+1, and so(10)m3,3Rn=5n−2d+1−2δRn−2−α1−4n+4d−4−2α2,1−α2,2+4δRn+2α1+τS1+τSs+∑i=2s−12τSi−μSi−α2,2−2α2,3−τS1−τSs−∑i=2s−12τSi−μSi−4α2,1−3α2,2−2α2,3+τS1+τSs+∑i=2s−14τSi−μSi+2νSi−∑i=2s−1τSi+νSi=n+2d−5+2δRn+α1−6α2,1−5α2,2−4α2,3+τS1+τSs+∑i=2s−13τSi−μSi+νSi.
The following results are a direct consequence of Theorem1.Corollary 1.
LetRn be a square-hexagonal chain containing d squares and n−d hexagons. Suppose that Rn consists of s segments S1,…,Ss and contains α1 kinks of type T1 and α2,j kinks of type T2,j, for j=1,2,3. Then,(11)BIDRn=∑2≤a≤b≤4ma,bRnθa,b=4θ2,3+θ3,3n+2θ3,3−4θ2,3d+2θ2,2+4θ2,3−5θ3,3+2θ2,2−4θ2,3+2θ3,3δRn+θ2,2−2θ2,3+θ3,3α1+2θ2,3−6θ3,3+4θ3,4α2,1+θ2,3−5θ3,3+θ2,4+3θ3,4α2,2+2θ2,4−4θ3,3+2θ3,4α2,3+θ3,3−θ2,3+θ2,4−θ3,4τS1+τSs+3θ3,3−2θ2,3+2θ2,4−4θ3,4+θ4,4∑i=2s−1τSi+θ2,3−θ3,3−θ2,4+θ3,4∑i=2s−1μSi+θ3,3−2θ3,4+θ4,4∑i=2s−1νSi.Corollary 2.
IfLn is a linear square-hexagonal chain with d squares and n−d hexagons, then(12)BIDLn=4θ2,3+θ3,3n+2θ3,3−4θ2,3d+2θ2,2+4θ2,3−5θ3,3+2θ2,2−4θ2,3+2θ3,3δLn.
Ln0 (resp. Ln1) denotes the linear square-hexagonal chain with n polygons where the terminal polygons are squares (resp. hexagons) and all nonterminal polygons are hexagons (resp. squares). Then,(13)BIDLn0=4θ2,3+θ3,3n+2θ2,2−4θ2,3−θ3,3,BIDLn1=3θ3,3n+6θ2,2+4θ2,3−5θ3,3.
The next theorem gives the extreme values of BID indices for the class of linear square-hexagonal chains.Theorem 2.
(a)
Ifθ3,3−2θ2,3>0, then(1)
BIDLn is minimum if and only if Ln≅Ln0;(2)
BIDLn is maximum if and only if Ln≅Ln1.(b)
Ifθ3,3−2θ2,3<0, then(1)
BIDLn is minimum if and only if Ln is linear polyomino chain;(2)
BIDLn is maximum if and only if Ln is linear hexagonal chain.Proof.
(a) Suppose thatθ3,3−2θ2,3>0. Let Ln be a linear square-hexagonal chain with d squares. Since d+δRn≥2 and δRn≥0, we obtain the following:(14)BIDLn=4θ2,3+θ3,3n+2θ3,3−4θ2,3d+δLn+2θ2,2+4θ2,3−5θ3,3+2θ2,2δLn≥4θ2,3+θ3,3n+22θ3,3−4θ2,3+2θ2,2+4θ2,3−5θ3,3=4θ2,3+θ3,3n+2θ2,2−4θ2,3−θ3,3=BIDLn0.
The equality holds if and only ifδLn=0 and d=2 equivalently Ln≅Ln0.
Also, we haved+δLn≤n. Therefore,(15)BIDLn=4θ2,3+θ3,3n+2θ3,3−4θ2,3d+2θ2,2+4θ2,3−5θ3,3+2θ2,2−4θ2,3+2θ3,3δLn=4θ2,3+θ3,3n+2θ3,3−4θ2,3d+δLn+2θ2,2+4θ2,3−5θ3,3+2θ2,2δLn≤4θ2,3+θ3,3n+2θ3,3−4θ2,3n+4θ2,3−5θ3,3+6θ2,2=3θ3,3n−5θ3,3+4θ2,3+6θ2,2=BIDLn1.
The equality holds if and only ifδLn=2 and d=n−2 equivalently Ln≅Ln1.
(b) is similar to the proof of part (a).
Now, we focus on the special cases of the first Zagreb indexM1θa,b=a+b and the second Zagreb index M2θa,b=ab of square-hexagonal chains.Corollary 3.
IfRn is a square-hexagonal chain with d squares, s segments S1,…,Ss, α1 kinks of type T1, and α2,j kinks of type T2,j, j=1,2,3,then(16)M1Rn=26n−8d+2α2,1+2α2,2+2α2,3−2,M2Rn=33n−6d+2δRn−13+α1+6α2,1+5α2,2+4α2,3−∑i=1sτSi−μSi−νSi.
The next result gives the extreme values of the first and second Zagreb indices for the class of square-hexagonal chains.Theorem 3.
(a)
M1Rn is minimum if and only if Rn is a linear polyomino chain.(b)
M1Rn is maximum if and only if Rn is a hexagonal chain.(c)
M2Rn is minimum if and only if Rn is a linear polyomino chain.(d)
M2Rn is maximum if and only if Rn is a zigzag hexagonal chain.Proof.
LetLPn denote the linear polyomino chain with n squares, Hn denote a hexagonal chain with n hexagons, and ZHn denote the zigzag hexagonal chain with n squares and n−2 kinks of type T1. Then, by Corollary 3, we have M1LPn=18n−2, M1Hn=26n−2, M2LPn=27n−13, and M2ZHn=34n−11. Let Rn be a square-hexagonal chain with d squares, s segments S1,…,Ss, and α1 kinks of type T1, α2,j kinks of type T2,j, j=1,2,3.(a)
Sinced≤n and α2,j≥0 for j=1,2,3, we have the following:(17)M1Rn=26n−8d+2α2,1+2α2,2+2α2,3−2≥18n−2=M1LPn,with equality holds if and onlyd=n and α2,1=α2,2=α2,3=0.(b)
Since all kinks of typeT2,j, j=1,2,3 are squares, we clearly see that d≥α2,1+α2,2+α2,3. Hence,(18)M1Rn=26n−24d−α2,1−α2,2−α2,3−2≤26n−2=M1Hn,and equality holds if and only ifd=0.(c)
IfSi is an internal segment and τSi=1, then each terminal polygon of Si is a kink of type T2,1 or type T2,2. This implies that ∑i=1sτSi≤α2,1+α2,2+1. Therefore,(19)M2Rn=33n−6d+2δRn−13+α1+6α2,1+5α2,2+4α2,3−∑i=1sτSi−μSi−νSi≥27n−13+6α2,1+5α2,2−∑i=1sτSi.Equality holds if and only ifd=n (consequently α2,2=0), δRn=0, and α2,3=∑i=1sμSi=∑i=1sνSi=0. Now, if α2,1≥1, then 6α2,1−∑i=1sτSi≥5, and hence M2Rn>27n−13=M2LPn. If α2,1=0, then ∑i=1sτSi=0, and then Rn≅LPn.(d)
We have the following:(20)M2Rn=33n−6d+2δRn−13+α1+6α2,1+5α2,2+4α2,3−∑i=1sτSi−μSi−νSi≤33n−13+2δRn−6d−α2,1−α2,2−α2,3−∑i=1sνSi−α2,2+α2,3−∑i=1sμSi+α1−α2,3−5∑i=1sνSi.
If, for somei, μSi≠0, then one of the terminal polygons of Si is a kink of type T2,2 or type T2,3. This implies that ∑i=1sμSi≤α2,2+α2,3. Moreover, if νSi≠0, then Si contains a square that is not a kink, and hence, d≥α2,3+α2,2+α2,3+∑i=1sνSi. On the other hand, we know that δRn≤2 and α1≤n−2. Thus, we have the following:(21)M2Rn≤34n−11=M2ZHn.
Now, if one of the valuesd, α2,1, α2,2, α2,3, ∑i=1sνSi, and ∑i=1sμSi is nonzero, then δRn≤1 or α1<n−2, and so M2Rn<34n−11=M2ZHn. Therefore, equality in (21) holds if and only if δRn=2 and α1=n−2 equivalently Rn≅ZHn.



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