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Extremal degree-based indices of general polyomino chains via dynamic programming

Manuel Montes-y-Morales, Sayle Sigarreta, Hugo Cruz-Suarez

TL;DR

A dynamic programming framework for identifying extremal general polyomino chains with respect to degree-based topological indices is developed and it is shown that the extremal configurations depend explicitly on the residue class of the number of squares modulo 4.

Abstract

In this paper, we develop a dynamic programming framework for identifying extremal general polyomino chains with respect to degree-based topological indices. As a concrete application, we resolve an open problem posed in 2015 by determining, for any given number of squares, the general polyomino chains that maximize the generalized Randić index with parameter $α=-1$. We show that the extremal configurations depend explicitly on the residue class of the number of squares modulo 4. Beyond this specific result, the proposed dynamic programming approach provides a constructive and systematic methodology for tackling extremal problems in graph theory.

Extremal degree-based indices of general polyomino chains via dynamic programming

TL;DR

A dynamic programming framework for identifying extremal general polyomino chains with respect to degree-based topological indices is developed and it is shown that the extremal configurations depend explicitly on the residue class of the number of squares modulo 4.

Abstract

In this paper, we develop a dynamic programming framework for identifying extremal general polyomino chains with respect to degree-based topological indices. As a concrete application, we resolve an open problem posed in 2015 by determining, for any given number of squares, the general polyomino chains that maximize the generalized Randić index with parameter . We show that the extremal configurations depend explicitly on the residue class of the number of squares modulo 4. Beyond this specific result, the proposed dynamic programming approach provides a constructive and systematic methodology for tackling extremal problems in graph theory.
Paper Structure (7 sections, 12 theorems, 39 equations, 6 figures, 4 algorithms)

This paper contains 7 sections, 12 theorems, 39 equations, 6 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Polyomino Chains and General Polyomino Chains
  3. General Polyomino Chains, Links and Actions
  4. Generating a General Polyomino Chain
  5. Dynamic Programming Approach Applied to General Polyomino Chains
  6. Extremal General Polyomino Chains for $R_{-1}$
  7. Final Discussion and Code

Key Result

Lemma 2.6

Let $(1,1,L_3,\dots,L_n)$ be a sequence of links, where $L_k\in\{1,2,3\}$ for each $k\geq 3$. If, at each construction step, the two ending vertices of the most recently added square have degree $2$, then the sequence is globally realizable.

Figures (6)

  • Figure 1: The eight restricted five-square chains (right/below rule) are not isomorphic to $PC_5$. Moreover, for $\alpha=-1$, their generalized Randić values are pairwise distinct.
  • Figure 5: Labeled $(i+1)$-th square.
  • Figure 6: Sequences of locally realizable links for the left and right graphs are $(1,1,1,2,1,1,1,3,1,1,1,1,3,1,3,1,1,2)$ and $(1,1,1,2,1,1,1,3,1,3,1,2,1,1,3,1,3,1)$, respectively. Both sequences correspond to the following sequence of actions $(SS, SC, CS, SS, SS, SC, CS, TT, CS, SC, CS, SS, SC, CS, TT, CS)$. The three initial squares are highlighted by a red dashed line.
  • Figure 7: Graphical representation of the three different safety zones.
  • Figure 8: According to Notation \ref{['n1']}, the graphs shown in (a), (b), (c), and (d) correspond, respectively, to $Z^{3}_{11}=C^{3}_{3}$, $C^{2,1}_{3,4}$, $\bar{C}^{3}_{3}$, and $\underaccent{\bar{}}{C}^{\,2,1}_{3,4}$.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 24 more