On the maximum positive semi-definite nullity and the cycle matroid of graphs

Abstract

Let G =( V,E )b e ag raph withV = {1,2,...,n}, in which we allow parallel edges but no loops, and let S+(G) be the set of all positive semi-definite n × n matrices A =( ai,j )w ith ai,j =0i fij and i and j are non-adjacent, ai,jfij and i and j are connected by exactly one edge, and ai,j ∈ R if i = j or i and j are connected by parallel edges. The maximum positive semi-definite nullity of G, denoted by M+(G), is the maximum nullity attained by any matrix A ∈S +(G). A k-separation of G is a pair of subgraphs (G1,G2) such that V (G1) ∪ V (G2 )= V , E(G1) ∪ E(G2 )= E, E(G1) ∩ E(G2 )= ∅ and |V (G1) ∩ V (G2)| = k .W henG has a k-separation (G1,G2 )w ithk ≤ 2, we give a formula for the maximum positive semi-definite nullity of G in terms of G1,G2 ,a nd in case ofk = 2, also two other specified graphs. For a graph G ,l etcG denote the number of components in G. As a corollary of the result on k-separations with k ≤ 2, we obtain that M+(G) − cG = M+(G � ) − cG for graphs G and Gthat have isomorphic cycle matroids.