On link-irregular graphs

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The subgraph of a graph G that is induced by the set of neighbors of a vertex v of G is the link of v. If every two distinct vertices of G have non-isomorphic links, then G is link-irregular. It is shown that there exists a link-irregular graph of order n if and only if n 6. The degree set D(G) of G is the set of degrees of the vertices of G. While there is no link-irregular graph G of order n such that |D(G)| {n, n -1}, it is shown that there exists a link-irregular graph G of order n such that |(G)| = n -2 if and only if n 7. Further, for each pair (d, n) of |(G)| = n -2 if and only if n 7. Further, for each pair (d, n) of integers with 3 d 8 and n d + 4, there is a link-irregular graph of order n whose degree set consists of n-d elements. The link-irregular ratio lir(G) of a link-irregular graph G is defined as |D(G)|/|V (G)|. For the set L of link-irregular graphs, it is shown that sup{ lir(G) : G L} = 1 and that 0 inf{ lir(G) : G L} 1/9.