The inertia set of a signed graph

Abstract

A signed graph is a pair (G, Σ), where G = (V, E) is a graph (in which parallel edges are permitted, but loops are not) with V = {1, . . . , n} and Σ ⊆ E. By S(G, Σ) we denote the set of all symmetric V × V matrices A = [ai,j] with ai,j < 0 if i and j are connected by only even edges, ai,j > 0 if i and j are connected by only odd edges, ai,j ∈ R if i and j are connected by both even and odd edges, ai,j = 0 if i = j and i and j are non-adjacent, and ai,i ∈ R for all vertices i. The stable inertia set of a signed graph (G, Σ) is the set of all pairs (p, q) for which there exists a matrix A ∈ S(G, Σ) with p positive and q negative eigenvalues which has the Strong Arnold Property. In this paper, we study the stable inertia set of (signed) graphs.

Signed graphs whose signed Colin de Verdière parameter is two

“…In [1] we also proved that a signed graph (G, Σ) has ν(G, Σ) ≤ 1 if and only if (G, Σ) is bipartite. For a positive integer n, we denote by K o n the signed graph (K n , E(K n )) and by K = n the signed graph (G, E(K n )), where G is the graph obtained from K n by adding to each edge an edge in parallel.…”

“…Then we obtain formulas which allow to calculate ν(G, Σ) from the values of ν on the parts of a 0-, 1-, 2-, or 3-split of (G, Σ). A proof of the following lemma can be found in [1].…”

“…In [1] we introduced for any signed graph (G, Σ) the signed graph parameter ν. In order to describe this parameter we need the notion of Strong Arnold Property (SAP for short).…”

“…A signed graph (G, Σ) has ν(G, Σ) ≤ 2 if and only if (G, Σ) has no minor isomorphic to K o 4 or K = 3 . One direction of the proof of this theorem follows easily from Theorem 1: As ν(K o 4 ) = ν(K = 3 ) = 3 (see [1]), any signed graph (G, Σ) with ν(G, Σ) ≤ 2 cannot have a minor isomorphic to K o 4 or K = 3 . The proof of the opposite direction spans the major part of this paper.…”

Using a new zero forcing process to guarantee the Strong Arnold Property

“…Remark 2.7. With and without the restriction of having the SAP, the inertia sets that can be achieved by matrices in S(G) are considered in the literature (e.g., see [1,6]). With the help of Theorem 2.6, if Z SAP (G) = 0, then these two inertia sets are the same.…”