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The notion of a $p$-Riordan graph generalizes that of a Riordan graph, which,in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. Inthis paper we introduce the notion of a $p$-Riordan word, and show how toencode $p$-Riordan graphs by $p$-Riordan words. For special important cases ofRiordan graphs (the case $p=2$) and oriented Riordan graphs (the case $p=3$) weprovide alternative encodings in terms of pattern-avoiding permutations andcertain balanced words, respectively. As a bi-product of our studies, weprovide an alternative proof of a known enumerative result on closed walks inthe cube.

Encoding labelled $p$-Riordan graphs by words and pattern-avoiding permutations