Abstract
Let be a graph. The atom-bond connectivity (ABC) index is defined as the sum of weights over all edges of , where denotes the degree of a vertex of . In this paper, we give the atom-bond connectivity index of the zigzag chain polyomino graphs. Meanwhile, we obtain the sharp upper bound on the atom-bond connectivity index of catacondensed polyomino graphs with squares and determine the corresponding extremal graphs.
1. Introduction
One of the most active fields of research in contemporary chemical graph theory is the study of topological indices (graph topological invariants) that can be used for describing and predicting physicochemical and pharmacological properties of organic compounds. In chemistry and for chemical graphs, these invariant numbers are known as the topological indices. There are many publications on the topological indices, see [1–6].
Let be a simple graph of order . A few years ago, Estrada et al. [7] introduced a further vertex-degree-based graph invariant, known as the atom-bond connectivity () index. It is defined as:
The index keeps the spirit of the Randić index, and it provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes [7]. Recently, the study of the index attracts some research attention [6, 8–12].
Polyomino graphs [13], also called chessboards [14] or square-cell configurations [15] have attracted some mathematicians’ considerable attention because many interesting combinatorial subjects are yielded from them such as domination problem and modeling problems of surface chemistry. A polyomino graph [16] is a connected geometric graph obtained by arranging congruent regular squares of side length 1 (called a cell) in a plane such that two squares are either disjoint or have a common edge. The polyomino graph has received considerable attentions.
Next, we introduce some graph definitions used in this paper.
Definition 1 (see [4]). Let be a polyomino graph. If all vertices of lie on its perimeter, then is said to be catacondensed polyomino graph or tree-like polyomino graph. (see Figure 1).
Definition 2 (see [16]). Let be a chain polyomino graph with squares. If the subgraph obtained from by deleting all the vertices of degree 2 and all the edges adjacent to the vertices is a path, then is said to be the zigzag chain polyomino graph, denoted by (see Figure 1).
In this paper, we give the indices of the zigzag chain polyomino graphs with squares and obtain the sharp upper bound on the indices of catacondensed polyomino graphs with squares and determine the corresponding extremal graphs.
2. The Indices of Catacondensed Polyomino Graphs
Let We call the weight of the edge , denoted by .
Note that for any catacondensed polyomino graph with squares, it can be obtained by gluing a new square to some catacondensed polyomino graph with squares. So, we have the following lemma.
Lemma 3. Let be a catacondensed polyomino graph with squares which is obtained by gluing a new square to some graph , where is a catacondensed polyomino graph with squares. One has (i), ,(ii), .
Proof. Consider the following: (i) if , by directly calculating, we have ,
(ii) now, let . Without the loss of generality, let square be adjacent to the edge in (see Figure 2). In the following, if the weights of some edges of have been changed when is adjacent to the edge in , then we marked these edges with thick lines in . Let . Note that except the edge of , the summation of the weights of the remaining three edges is always in . There are exactly three types of formations (see Figure 2).
Case 1. In Type I, and (see Figure 3).
By the definition of index, we have = + + + = (). If , then . If or , then . If , then .Case 2. In Type II, , ,, and. (see Figure 4).
Let adjacent to and , adjacent to (see Figure 4). Then , , and . If , , and , which is in contradiction with ; if , which is in contradiction with ; if , which is in contradiction with ().
By the definition of index, we have = + + + + = (). If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then .
Case 3. In Type III, and (see Figure 5).
Let , adjacent to and , adjacent to (see Figure 5). Then, , , , and . Since the case is the same as , where . And note that if or , the vertices and are symmetric.
By the definition of index, we have (). If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then . If , then .
By directly calculating, we have , , , , , , , and , . So , where .
Therefore, .
By Lemma 3, we have the following theorem.
Theorem 4. Let be a catacondensed polyomino graph with squares, then where is a nonnegative integer for and .
Proof. We prove Theorem 4 by the induction on . If , by directly calculating, we have , where (). So, Theorem 4 holds for .
Assume that Theorem 4 holds for all catacondensed polyomino graphs with () squares, that is,
where is a nonnegative integer for and .
We will prove that Theorem 4 holds for in the following. Let be a catacondensed polyomino graph with squares. Without the loss of generality, can be obtained from some catacondensed polyomino graph with squares by gluing a new square to . By Lemma 3, we have . It means that , where . By the induction assumption and direct computation, we have
There exists some such that and for (). Obviously, is a nonnegative integer for and .
Lemma 5. Let be a catacondensed polyomino graph with squares. If , there are exactly four nonisomorphism catacondensed polyomino graphs (see Figure 6), where , , , .
Theorem 6. Let be a zigzag chain polyomino graph with squares, then
Proof. Obviously, can be obtained by gluing a new square to . Let be the square adjacent to (see Figure 1). We will prove Theorem 6 by the induction on .
If , then Theorem 6 holds (by Lemma 5). Assume that for . By the induction assumption and the in Lemma 3, we have
So, Theorem 6 holds.
Note that for and by Lemma 5, we obtain the following Theorem 7.
Theorem 7. Let be a catacondensed polyomino graph with squares, then , with the equality if and only if .
Acknowledgments
The project was supported by NSFC (no. 11071272; 11101086; 11101087) and the Foundation of Fuzhou University (no. 2012-XQ-30).