Abstract
Polyomino graphs is one of the research objectives in statistical physics and in modeling problems of surface chemistry. A random polyomino chain is a subgraph of a polyomino graph. The matching energy is defined as the sum of the absolute values of the zeros of the matching polynomial of a graph. In this paper, we characterize the graphs with the extremal matching energy among all random polyomino chains of a polyomino graph by the probability method.
Highlights
A polyomino graph [1] is a connected geometric graph obtained by arranging congruent regular squares of side length 1 in a plane such that two squares are either disjoint or have a common edge
Considerable research in statistical physics and structural chemistry has been devoted to polyomino graphs [4,5,6,7,8,9,10,11,12,13,14]
A polyomino chain Qn with n squares, which is a subgraph of a polyomino graph, can be regarded as a polyomino chain Qn−1 with n − 1 squares adjoining to a new terminal square by a cut edge, see Figure 1
Summary
A polyomino graph [1] ( called chessboards [2] or square-cell configurations [3]) 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. A polyomino chain Qn with n squares, which is a subgraph of a polyomino graph, can be regarded as a polyomino chain Qn−1 with n − 1 squares adjoining to a new terminal square by a cut edge, see Figure 1. Gutman and Wagner [19] first proposed the concept of the matching energy of a graph, denoted by ME( G ), as ME( G ) =. For a random polyomino chain Q(n, t), the matching energy is a random variable. We shall determine the extremal graphs with respect to the matching energy for all random polyomino chains
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