Terwilliger graphs in which the neighborhood of some vertex is isomorphic to a Petersen graph

A study was conducted to analyze the Terwilliger graphs in which the neighborhood of some vertex is isomorphic to a Peterson graph. A graph is called biregular if the degrees of its vertices take exactly two values, while the subgraph is called μ-subgraph if the vertices are a distant of 2 apart. A clique extension of the graph obtained by replacing each vertex with a clique such that the various cliques are pairwise disjoint and a vertex is adjacent to a vertex. The class of strongly regular graphs is denoted as μ = 1 in which the neighborhood of any vertex is the union of isolated cliques of particular order. The Terwilliger graph is an incomplete connected graph, in which the subgraph on any vertices is a μ-clique, where μ is a constant. The n-hedgehog is a graph obtained from an n-clique by adding an n-coclique such that each of its vertices is adjacent to exactly one vertex of the clique and each vertex of the clique is adjacent to exactly one vertex of the coclique.