On-Bond Incident Degree Indices of Square-Hexagonal Chains

For a graph G , its bond incident degree (BID) index is defined as the sum of the contributions f ( d u ,d v ) over all edges uv of G , where d w denotes the degree of a vertex w of G and f is a real-valued symmetric function. If f ( d u , d v ) (cid:29) d u + d v or d u d v , then the corresponding BID index is known as the first Zagreb index M 1 or the second Zagreb index M 2 , 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 M 1 and M 2 are also characterized from the class of all square-hexagonal chains having a fixed number of polygons.


Introduction
ose (chemical) graph-theoretical terminologies and notations adopted in the current paper that are not de ned here in this paper can be found in some standard (chemical) graph-theoretical books; for example, [1][2][3]. All the graphs to be considered in the current paper are nite and connected.
In what follows, it is assumed that G is a graph. e edge set and vertex set of G are denoted by E(G) and V(G), respectively. For a vertex w ∈ V(G), its degree is denoted by d w (G) (or simply by d w 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. e rst and second Zagreb indices [4], appeared in the rst 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 M 1 and M 2 , respectively, and for G, they are de ned as follows: It is known that u∈V(G) (d u ) 2 uv∈E(G) (d u + d v ). Most of their known properties can be found in the review paper [4] and in the related references included therein.
For G, its bond incident degree (BID) index is de ned as follows: (2) where is the degree of the vertex u, f is a real-valued function such that f(d u , d v ) f(d v , d u ), uv is the edge d u with end vertices u and v of G, Δ(G) is the maximum degree in G, θ a,b f(a, b) and m a,b (G) 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 θ a,b a + b or θ a,b ab, then the corresponding BID index is M 1 or M 2 , respectively. Details about some mathematical aspects of the BID indices can be found in the papers [6][7][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 graph ID 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. R n refers to a square-hexagonal chain consisting of n polygons. If all polygons in R n are hexagons, then we say that R n is a hexagonal chain (see, for instance, [9]) and if all the polygons are squares, then R n is known as a polyomino chain (see, for instance, [10]). Also, if squares and hexagons are concatenated alternately in R n , then we say that R n 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]. e present study can be considered as a continuation of the research conducted in [10,12] and it is motivated from the paper [13][14][15] concerning squarehexagonal chains. In the current paper, a general expression for calculating any BID index of square-hexagonal chains is derived. e chains attaining the maximum or minimum values of M 1 and M 2 are also characterized from the class of all square-hexagonal chains having a fixed number of polygons.

Main Results
In order to obtain the main results, we require some terminology concerning square-hexagonal chains. In a squarehexagonal 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 ( Figure 1) 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 squarehexagonal chains that contain the following types of kinks: (1) Kinks of type T 1 : A nonterminal hexagon having exactly two adjacent vertices of degree two (see Figure 1); (2) Kinks of type T 2,1 : A nonterminal square containing a vertex of degree two and adjacent to two squares (see Figure 2(a)); (3) Kinks of type T 2,2 : A nonterminal square containing a vertex of degree two and adjacent to a square and a hexagon (see Figure 2(b)); (4) Kinks of type T 2,3 : A nonterminal square containing a vertex of degree two and adjacent to two hexagons (see Figure 2(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. e length l(S) of a segment S is its number of polygons. E(S) 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 squarehexagonal chain consists of s segments if and only if it contains exactly s − 1 kinks.
For a square-hexagonal R n , we define the value δ R n to be the number of terminal hexagons in R n . We also define the following values for segments S 1 , . . . , S s of R n : see Now, we are ready to establish the general expression for calculating the BID indices of square-hexagonal chains. Theorem 1. Let R n be square-hexagonal chain containing d squares and n − d hexagons. Suppose that R n consists of s segments S 1 , . . . , S s and contains α 1 kinks of type T 1 and α 2,j kinks of type T 2,j , for j � 1, 2, 3. en,

Journal of Mathematics
Proof. Let C 1 , . . . , C s−1 be the kinks of R n such that C i joins segments S i and S i+1 .
Clearly, the collection A 1 , . . . , A s−1 , B 1 , . . . , B s forms a partition for E(R n ) . For 2 ≤ a ≤ b ≤ 4, i � 1, . . . , s − 1,  j � 1, . . . , s, let p (i) a,b be the number of edges in A i that connects vertices of degrees a and b and q (j) a,b be the number of edges in B j that connect vertices of degrees a and b. en, a,b . First, we calculate m 2,2 (R n ). We have q (1) 2,2 + q (s) 2,2 � 2 (δ R n + 1), and for j � 2, . . . , s − 1, we have q (j) 2,2 � 0. Also, a kink contains an edge joining vertices of degree 2 if and only if it is of type T 1 . Hence, m 2,2 � 2δ R n + α 1 + 2.
To calculate m 4,4 (R n ), note that a segment S i contains an edge joining vertex of degree 4 if and only if S i is an internal segment and τ(S i ) � 1 or ](S i ) � 1. us, m 4,4 (R n ) � s−1 i�2 (τ(S i ) + ](S i )).

(7)
Substituting the values of m 2,4 (R n ) and m 4,4 (R n ) in (7) and solving the resulting equation for m 3,4 (R n ) yield the following: 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 type T 2,1 , T 2,2 , or T 2,3 . Now, adding number of vertices of degree 2 in the terminal polygons, we see that the number of vertices of degree 2 in R n is 2n − 2d + 4 + α 2,1 + α 2,2 + α 2,3 . Hence, e total number of edges in R n is n + 1 + 2 d + 4(n− d) � 5n − 2 d + 1, and so e following results are a direct consequence of eorem 1. □ Corollary 1. Let R n be a square-hexagonal chain containing d squares and n − d hexagons. Suppose that R n consists of s segments S 1 , . . . , S s and contains α 1 kinks of type T 1 and α 2,j kinks of type T 2,j , for j � 1, 2, 3. en,

(13)
The next theorem gives the extreme values of BID indices for the class of linear square-hexagonal chains. Proof. (a) Suppose that θ 3,3 − 2θ 2,3 > 0. Let L n be a linear square-hexagonal chain with d squares. Since d + δ R n ≥ 2 and δ R n ≥ 0, we obtain the following: e equality holds if and only if δ L n � 0 and d � 2 equivalently L n � L 0 n .
Journal of Mathematics e equality holds if and only if δ L n � 2 and d � n − 2 equivalently L n � L 1 n . (b) is similar to the proof of part (a). Now, we focus on the special cases of the first Zagreb index M 1 (θ a,b � a + b) and the second Zagreb index M 2 (θ a,b � ab) of square-hexagonal chains. □ Corollary 3. If R n is a square-hexagonal chain with d squares, s segments S 1 , . . . , S s , α 1 kinks of type T 1 , and α 2,j kinks of type T 2,j , j � 1, 2, 3,then The next result gives the extreme values of the first and second Zagreb indices for the class of square-hexagonal chains. Proof. Let LP n denote the linear polyomino chain with n squares, H n denote a hexagonal chain with n hexagons, and ZH n denote the zigzag hexagonal chain with n squares and n − 2 kinks of type T 1 . en, by Corollary 3, we have M 1 (LP n ) � 18n − 2, M 1 (H n ) � 26n − 2, M 2 (LP n ) � 27 n − 13, and M 2 (ZH n ) � 34n − 11. Let R n be a square-hexagonal chain with d squares, s segments S 1 , . . . , S s , and α 1 kinks of type T 1 , α 2,j kinks of type T 2,j , j � 1, 2, 3.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.