Tensors and Graphs

Abstract

In § 1, we discuss symmetry classes of tensors and their dimensions in the context of the epresentation theory of the general linear group. The main result is a formula for the Marcus–Chollet index. In § 2, we observe that the free vector space generated by the nonisomorphic graphs on p vertices is a symmetry class of tensors. Thus, we are able to make use of the dimension formulas in § 1 to enumerate the nonisomorphic graphs. For example: let $m = p ( p - 1 ) /2$. Denote by $\xi _q $ the irreducible character of $S_m $ corresponding to the partition $( m - q,q )$, $0\leqq q\leqq m/2$. Then the number of nonisomorphic, unlabelled graphs on p vertices is $\sum_{q = 0}^{[ m/2 ]} ( m - 2q + 1 ) ( 1,\xi_q )_p $, where $( 1,\xi _q )_p $ is the number of occurrences of the principal character in the restriction of $\xi _q $ to the pair group $S_p^{( 2 )} $ (i.e., the line group of the complete graph on p vertices). In addition, we use the corresponding representation of the general linear group to catalog distance inventories between graphs. Section 3 contains extensions to multigraphs, and § 4 some combinatorial lemmas and proofs.