Two-Dimensional Digitized Picture Arrays and Parikh Matrices

Parikh matrix mapping or Parikh matrix of a word has been introduced in the literature to count the scattered subwords in the word. Several properties of a Parikh matrix have been extensively investigated. A picture array is a two-dimensional connected digitized rectangular array consisting of a finite number of pixels with each pixel in a cell having a label from a finite alphabet. Here we extend the notion of Parikh matrix of a word to a picture array and associate with it two kinds of Parikh matrices, called row Parikh matrix and column Parikh matrix. Two picture arrays A and B are defined to be M-equivalent if their row Parikh matrices are the same and their column Parikh matrices are the same. This enables to extend the notion of M-ambiguity to a picture array. In the binary and ternary cases, conditions that ensure M-ambiguity are then obtained

Algebraic Properties of Parikh Matrices of Words under an Extension of Thue Morphism

The Parikh matrix of a word $w$ over an alphabet $\{a_1, \cdots , a_k \}$ with an ordering $a_1 < a_2 < \cdots a_k,$ gives the number of occurrences of each factor of the word $a_1 \cdots a_k$ as a (scattered) subword of the word $w.$ Two words $u,v$ are said to be $M-$equivalent, if the Parikh matrices of $u$ and $v$ are the same. On the other hand properties of image words under different morphisms have been studied in the context of subwords and Parikh matrices. Here an extension to three letters, introduced by S$\acute{e}\acute{e}$bold (2003), of the well-known Thue morphism on two letters, is considered and properties of Parikh matrices of morphic images of words are investigated. The significance of the contribution is that various classes of binary words are obtained whose images are $M-$equivalent under this extended morphism

Algebraic Properties of Parikh Matrices of Binary Picture Arrays

A word is a finite sequence of symbols. Parikh matrix of a word is an upper triangular matrix with ones in the main diagonal and non-negative integers above the main diagonal which are counts of certain scattered subwords in the word. On the other hand a picture array, which is a rectangular arrangement of symbols, is an extension of the notion of word to two dimensions. Parikh matrices associated with a picture array have been introduced and their properties have been studied. Here we obtain certain algebraic properties of Parikh matrices of binary picture arrays based on the notions of power, fairness and a restricted shuffle operator extending the corresponding notions studied in the case of words. We also obtain properties of Parikh matrices of arrays formed by certain geometric operations

Certain Distance-Based Topological Indices of Parikh Word Representable Graphs

Relating graph structures with words which are finite sequences of symbols, Parikh word representable graphs ($PWRGs$) were introduced. On the other hand in chemical graph theory, graphs have been associated with molecular structures. Also several topological indices have been defined in terms of graph parameters and studied for different classes of graphs. In this paper, we derive expressions for computing certain topological indices of $PWRGs$ of binary core words, thereby enriching the study of $PWRGs.

Parikh Matrices and Istrail Morphism

A word w is a sequence of symbols. A scattered subword or simply a subword u of the word w is a subsequence of w. Parikh matrix M(w) is an ingenius tool introduced by Mateescu et al (2001) to count certain subwords in a word w. Various properties of Parikh matrices have been established. Two words u and v are said to be M-ambiguous or amiable if their Parikh matrices M(u) and M(v) are the same. On the other hand a morphism f is a mapping on words w whose images f(w) are also words with the property that, f(uv)=f(u)f(v) for given words u and v. Istrail morphism (Istrail, 1977) is a specific kind of morphism on a set {a,b,c} of three symbols. Using this morphism, M-ambiguity or amiability of words based on Parikh matrices is investigated by Atanasiu (2010). Parikh matrices of words that involve certain ratio-property are investigated by Subramanian et al (2009). Here we consider this kind of ratio-property in the context of Istrail morphism and obtain certain properties of morphic images of words under Istrail morphism. Using these properties, conditions are obtained for product of Parikh matrices of such morphic images under Istrail morphism to commute