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 polyo...

It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino that do not contain a northeast chain of size k depends only on the set of columns of the polyomino, but not the shape of the polyomino. Rubey’s proof is an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes. In this paper we present a bijective proof for this result ...

In this paper we give simple bijective proofs that the number of fillings of layer polyominoes with no northeast chains is the same as the number with no southeast chains. We consider 01fillings and N-fillings, and prove the results for both strong chains where the smallest rectangle containing the chain is also in the polyomino, and for regular chains where only the corners of the smallest rec...

We show that the zig-zag chain Z3 n of segments of length 3 (see Figure 2) has the minimal ABC index among all polyomino chains with n squares. More generally, we give conditions on the numbers {φij} under which the zig-zag chain Z3 n is an extremal value of the induced topological index T defined by T (G) = ∑ 1≤i≤j≤n−1 mijφij where G is a graph with n vertices and mij is the number of edges of...

In polyomino achievement games, two players alternately mark the cells of a tessellation and try to achieve a given polyomino. In [2], Bode and Harborth investigated polyomino achievement games for the hexagonal tessellation and determined all but five polyominoes with at most five cells whether they are achieved by the first player. In this paper, we show winning strategies for three hexagonal...

The site-perimeter of a polyomino is the number of empty cells connected to the polyomino by an edge. A formula for the minimum site-perimeter with a given cell size is found. This formula is used to show the effectiveness of a simple random strategy in polyomino set achievement games.

In theoretical chemistry, the researchers use graph models to express the structure of molecular, and the Zagreb indices and multiplicative Zagreb indices defined on molecular graph G are applied to measure the chemical characteristics of compounds and drugs. In this paper, we present the exact expressions of multiplicative Zagreb indices for certain important chemical structures like nanotube,...

Several topological indices are known to have widespread implications in a variety of research areas. Over the years, Kirchhoff index has turned out be an extremely significant and efficient index. In this paper, we propose exact formulas for expected values random polyomino chain construct multiplicative degree-Kirchhoff additive We also carefully examine highest degree through

The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $chi_{alpha}$ values from a certain collection of polyomino ...

the present note is devoted to establish some extremal results for the zeroth-order general randi'{c} index of cacti, characterize the extremal polyomino chains with respect to the aforementioned index, and hence to generalize two already reported results.