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A graph $G=(V,E)$ is word-representable if and only if there exists a word$w$ over the alphabet $V$ such that letters $x$ and $y$, $x\neq y$, alternatein $w$ if and only if $xy\in E$. A split graph is a graph in which the verticescan be partitioned into a clique and an independent set. There is a long lineof research on word-representable graphs in the literature, and recently,word-representability of split graphs has attracted interest.  In this paper, we first give a characterization of word-representable splitgraphs in terms of permutations of columns of the adjacency matrices. Then, wefocus on the study of word-representability of split graphs obtained byiterations of a morphism, the notion coming from combinatorics on words. Weprove a number of general theorems and provide a complete classification in thecase of morphisms defined by $2\times 2$ matrices.

Word-representability of split graphs generated by morphisms