Abstract

For a graph , its bond incident degree (BID) index is defined as the sum of the contributions over all edges of , where denotes the degree of a vertex of and is a real-valued symmetric function. If or , then the corresponding BID index is known as the first Zagreb index or the second Zagreb index , 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 and are also characterized from the class of all square-hexagonal chains having a fixed number of polygons.

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 that is a graph. The edge set and vertex set of are denoted by and , respectively. For a vertex , its degree is denoted by (or simply by 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 and , respectively, and for , they are defined as follows:

It is known that . Most of their known properties can be found in the review paper [4] and in the related references included therein.

For , its bond incident degree (BID) index is defined as follows:where is the degree of the vertex , is a real-valued function such that , is the edge with end vertices and of , is the maximum degree in , and is the number of those edges of whose one end vertex has the degree and the other end vertex has the degree . We note here that if or , then the corresponding BID index is or , 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 graph whose vertices are the polygons of the considered square-hexagonal system, while there is an edge between two vertices of 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. refers to a square-hexagonal chain consisting of polygons. If all polygons in are hexagons, then we say that is a hexagonal chain (see, for instance, [9]) and if all the polygons are squares, then is known as a polyomino chain (see, for instance, [10]). Also, if squares and hexagons are concatenated alternately in , then we say that 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 and 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 (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 square-hexagonal chains that contain the following types of kinks:(1)Kinks of type : A nonterminal hexagon having exactly two adjacent vertices of degree two (see Figure 1);(2)Kinks of type : A nonterminal square containing a vertex of degree two and adjacent to two squares (see Figure 2(a));(3)Kinks of type : 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 : A nonterminal square containing a vertex of degree two and adjacent to two hexagons (see Figure 2(c));

Figure 1 

Kinks of type .

(a)

(b)

(c)

(a)
(b)
(c)

Figure 2 

(a) Kinks of type ; (b) Kinks of type ; (c) Kinks of type .

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 length of a segment is its number of polygons. refers to the set of all edges of a segment . 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 segments if and only if it contains exactly kinks.

For a square-hexagonal , we define the value to be the number of terminal hexagons in . We also define the following values for segments of : see Figures 3 and 4.

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, .

Now, we are ready to establish the general expression for calculating the BID indices of square-hexagonal chains.

Theorem 1. Let be square-hexagonal chain containing squares and hexagons. Suppose that consists of segments and contains kinks of type and kinks of type , for . Then,

Proof. Let be the kinks of such that joins segments and .
Let , and , for . Also, we set , , and , for .
Clearly, the collection forms a partition for . For , , , let be the number of edges in that connects vertices of degrees and and be the number of edges in that connect vertices of degrees and . Then, we get .
First, we calculate . We have , and for , we have . Also, a kink contains an edge joining vertices of degree 2 if and only if it is of type . Hence, .
To calculate , note that a segment contains an edge joining vertex of degree 4 if and only if is an internal segment and or . Thus, .
Next, we calculate . Vertices of degree 4 appear only in kinks of type , , and . In fact, , , and for , if is a kink of type , , or , and otherwise. Hence, . Also, . Thus,Now, each kink of types , , or contains exactly one vertex of degree 4, and so the number of vertices of degree 4 is . Therefore,Substituting the values of and in (7) and solving the resulting equation for 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 , , or . Now, adding number of vertices of degree 2 in the terminal polygons, we see that the number of vertices of degree 2 in is . Hence, . Therefore,The total number of edges in is , and soThe following results are a direct consequence of Theorem 1.

Corollary 1. Let be a square-hexagonal chain containing squares and hexagons. Suppose that consists of segments and contains kinks of type and kinks of type , for . Then,

Corollary 2. If is a linear square-hexagonal chain with squares and hexagons, then (resp. ) denotes the linear square-hexagonal chain with polygons where the terminal polygons are squares (resp. hexagons) and all nonterminal polygons are hexagons (resp. squares). Then,The next theorem gives the extreme values of BID indices for the class of linear square-hexagonal chains.

Theorem 2. (a)If , then(1) is minimum if and only if ;(2) is maximum if and only if .(b)If , then(1) is minimum if and only if is linear polyomino chain;(2) is maximum if and only if is linear hexagonal chain.

Proof. (a) Suppose that . Let be a linear square-hexagonal chain with squares. Since and , we obtain the following:The equality holds if and only if and equivalently .
Also, we have . Therefore,The equality holds if and only if and equivalently .
(b) is similar to the proof of part (a).
Now, we focus on the special cases of the first Zagreb index and the second Zagreb index of square-hexagonal chains.