1 Structural properties of word representable graphs Somnath Bera and Kalpana Mahalingam 1 Keywords. Parikh matrices, Parikh word representable graph, bipartite chordal graph, graph iso- morphism, Hamiltonian cycle. Abstract. Given a word w = w1w2...wn of length n over an ordered alphabet Σ k , we construct a graph G(w)=(V (w),E(w)) such that V (w) has n vertices labeled 1, 2, ··· ,n and for i, j ∈ V (w), (i, j ) ∈ E(w) if and only if wi wj is a scattered subword of w of the form at at+1, at ∈ Σ k , for some 1 ≤ t ≤ k - 1 with the ordering at <at+1. A graph is said to be Parikh word representable if there exists a word w over Σ k such that G = G(w). In this paper we characterize all Parikh word representable graphs over the binary alphabet in terms of chordal bipartite graphs. It is well known that the graph isomorphism (GI) problem for chordal bipartite graph is GI complete. The GI problem for a subclass of (6, 2) chordal bipartite graphs has been addressed. The notion of graph powers is a well studied topic in graph theory and its applications. We also investigate a bipartite analogue of graph powers of Parikh word representable graphs. In fact we show that for G(w), G(w) [3] is a complete bipartite graph, for any word w over binary alphabet. 1. Introduction The Parikh vector mapping, an important tool in the theory of formal languages, introduced by R. J. Parikh in [7] - gives the number of occurrences of letters in the word as a numerical vector. After a long gap the Parikh matrix of a word has been introduced in [1] as an extension of Parikh vector mapping. The Parikh matrix mapping of a word gives more numerical properties of a word in terms of certain subwords of the given word. The Parikh vector appears in the Parikh matrix as the second upper diagonal. Hence two words having same the Parikh vector need not have the same Parikh matrix; in other words this mapping is not injective. Two words α and β are said to be amiable if and only if they have the same Parikh matrix [2]. In [2] the author had constructed a graph with vertex set being the set of all amiable words over binary alphabet and have shown that such a graph is connected. Another type of word representable graphs has its roots in the study of Perkins semigroup [3, 4]. A graph G =(V,E) is word representable [3, 4] if there exists a word w over the alphabet Σ such that letters x and y alternate in w if and only if (x, y) ∈ E for each x 6= y. In this paper we introduce another approach using Parikh matrices to represent graphs with words. Let Σ k = {a i |1 ≤ i ≤ k} be an alphabet with an ordering a i <a i+1 , 1 ≤ i ≤ k - 1.A graph G =(V,E) is called Parikh word representable or simply word representable if there exists a word w = w 1 w 2 ...w n ∈ Σ * k of length n such that V is the set of vertices {1, 2, ··· ,n} and for i, j ∈ V , (i, j ) ∈ E iff w i w j is a scattered subword of w of the form a t a t+1 , for some 1 ≤ t ≤ k - 1. In this paper we characterize the class of all graphs that are word representable over an ordered binary alphabet. We show that these graphs are indeed (6, 2) chordal bipartite graphs with an addi- tional property pertaining to the degree of the vertices. A graph G is called (6, 2) chordal bipartite if G is bipartite and for each cycle of length at least 6 there exists at least 2 chords. We also show that the class of word representable graphs is a proper subclass of bipartite permutation graphs. 1 Corresponding author