SIGNED GRAPHS WITH TOTALLY DISCONNECTED STAR COMPLEMENTS

. We are interested in a signed graph ˙ G which admits a decomposition into a totally disconnected (i.e., without edges) star complement and a signed graph ˙ S induced by the star set. In this study we derive certain properties of ˙ G ; for example, we prove that the number of (distinct) eigenvalues of ˙ S does not exceed the number of those of ˙ G . Some particular cases are also considered.


Introduction
Given a graph G = (V (G), E(G)), let σ : E(G) → {1, −1}. ThenĠ = (G, σ) is a signed graph derived from its underlying graph G. To ease language, we say that each edge e ∈ E(G) is accompanied by the sign σ(e) ∈ {1, −1}. The (multiplicative) group {1, −1} can also be written as {+, −}. We denote |V (Ġ)| by n. The edge set of a signed graph is composed of subsets containing positive and negative edges, respectively. Throughout the paper, we interpret an (unsigned) graph as a signed graph with all the edges being positive.
The adjacency matrix AĠ ofĠ is obtained from the (0, 1)-adjacency matrix of G by reversing the sign of all 1s which correspond to negative edges. The eigenvalues of AĠ, together with their repetition, form the spectrum ofĠ.
We recall from [3,6] the existence of a spectral tool (known as a star complement technique) for constructing comparatively large graphs with specified spectral properties from their smaller parts, known as star complements; these can be defined in the case of signed graphs as well (the definition is given in the next section). This motivates us to consider signed graphs which admit a vertex set partition into two non-empty subsets such that the subgraph induced by one of these subsets is a particular (totally disconnected) star complement inĠ. IfĠ admits the mentioned partition, then we briefly say thatĠ can be decomposed into the corresponding 96 ZORAN STANIĆ subgraphs. We derive certain properties of such signed graphs and their star complements, and consider some particular cases. Similar decompositions of graphs can be found in [5,10].

Preliminaries
Our terminology and notation are transferred from the domain of graphs-see [2,3,8]; for example, we say that a signed graph is connected, regular, or bipartite if the same holds for its underlying graph. We use K n to denote the complete graph with n vertices and kK n for the union of k disjoint copies of K n . In particular, nK 1 denotes the totally disconnected (i.e., without edges) graph with n vertices. The all-1, the all-0, and the identity matrices are denoted by J, O, and I, respectively. To avoid possible confusion, their size will occasionally be given as a subscript.
If the vertices i and j are adjacent (i.e., if there is either a positive or a negative edge between them), we write i ∼ j. In particular, i + ∼ j (resp. i − ∼ j) designates the existence of a positive (resp. negative) edge between i and j.
The graphsĠ andḢ are said to be switching equivalent if there exists a set U ⊆ V (Ġ) such thatḢ is obtained fromĠ by reversing the sign of every edge with exactly one end in U . Switching equivalent signed graphs share the same underlying graph and the same spectrum. Equivalently, if the vertex labelling ofĠ andḢ is transferred from their common underlying graph, then they are switching equivalent if there exists a diagonal matrix D of ±1s such that AḢ = D −1 AĠD. We say thatĠ andḢ are switching isomorphic if one of them is isomorphic to a signed graph which is switching equivalent to the other one. Switching isomorphic signed graphs are usually identified.
A cycle in a signed graph is said to be positive if the number of negative edges contained is not odd. Otherwise, it is said to be negative.
A signed doubled graphG is obtained by inserting a negative (parallel) edge between every pair of adjacent vertices of a graph G; precisely,G is a signed multigraph.
We proceed by a concept of signed line graphs which is frequently used in spectral considerations: see [1,4,7,9]; a different approach can be found in [11]. For a signed graphĠ, we introduce the vertex-edge orientation η : V (Ġ) × E(Ġ) → {1, 0, −1} formed by obeying the following rules: In fact, each edge gets two orientations, so η is also called a bi-orientation. The vertex-edge incidence matrix B η is the matrix whose rows and columns are indexed by V (Ġ) and E(Ġ) respectively, such that its (i, e)-entry is equal to η(i, e). The following relation, valid also in the case that multiple edges exist, is well known: where L(Ġ) is taken to be the signed line graph ofĠ. A signed line graph defined in this way depends on orientation (in other words, it is not unique), but it is easy to show that different orientations produce switching equivalent signed line graphs. We also observe that this concept does not generalize the concept of line graphs.
For example, the signed line graph of the positive triangle is the negative triangle, while the triangle (interpreted as a graph) is isomorphic to its line graph.
Since B η B η is positive semidefinite, we deduce that the spectrum of L(Ġ) is bounded (from below) by −2. In analogy to the theory of unsigned graphs, we say that a connected signed graph is exceptional if its least eigenvalue is greater than or equal to −2, but it is not a signed line graph. All the exceptional signed graphs whose least eigenvalue is greater than −2 are obtained in [4]. Up to switching isomorphism, there are exactly 1507 exceptional signed graphs with the above property; each of them has six, seven, or eight vertices.
We continue with star complements. The entire concept is transferred from the domain of graphs [3,6]. Given a signed graphĠ with n vertices, let µ be its eigenvalue and P a matrix representing the orthogonal projection of R n onto the eigenspace E(µ) with respect to the canonical basis {e 1 , e 2 , . . . , e n }. The set S ⊆ V (Ġ) = {1, 2, . . . , n} such that the vectors P e i (i ∈ S) make a basis of E(µ) is called a star set for µ. We useṠ to denote the signed graph induced by S. The signed graph induced by the set V (Ġ) \ S is called a star complement for µ. It follows that µ is not an eigenvalue of the star complement and |S| is equal to the multiplicity of µ inĠ. We use the notation k = n − |S|.
The following result is known as the Reconstruction Theorem.
This result was initially formulated in the context of graphs. Since every graph can be interpreted as a signed graph, it follows that Theorem 2.1 is an extension to the field of signed graphs.
IfḢ is a star complement for an eigenvalue µ and x, y ∈ R k , we define the bilinear form By virtue of Theorem 2.1, if µ is not an eigenvalue ofḢ (with k vertices), then there exists a signed graphĠ withḢ as a star complement for µ if and only if for all distinct i, j ∈ S, where b i and b j determine neighbourhoods of i and j inḢ, respectively.

ZORAN STANIĆ
Each of the mentioned 1507 exceptional signed graphs is a star complement for −2 in an (also) exceptional signed graph. Thus, we can say that exceptional signed graphs are characterized by means of their star complements. A similar characterization of exceptional graphs can be found in [3]. Here is another illustration.

Example 2.2.
A triangle with positive edges is a star complement for −2 in the signed graph of Figure 1 (with spectrum 2 3 , (−2) 3 ). Indeed, by using Theorem 2.1, we easily compute the matrix B and the adjacency matrix ofṠ. Observe that the resulting signed graph is not exceptional.

Results
We start with the following theorem.
(ii): Considering the corresponding top-left block of AĠ, we immediately get X (1) = AṠ and X (2) (iii): Assume thatĠ has exactly t eigenvalues. Since the top-left block of A pĠ is a polynomial in AṠ of degree at most p (by (3.1)), considering the minimal polynomial of AĠ we deduce that there exists a polynomial P of degree at most t such that P (AṠ) = O, which concludes the proof.
Assume now thatĠ has two eigenvalues, λ and µ. Computing its minimal polynomial, we get It follows immediately thatĠ is regular with vertex degree −λµ, which implies that λ and µ are non-zero; in fact, exactly one of them is positive.

Lemma 3.2.
If kK 1 is a star complement for µ in a signed graphĠ with n vertices and two eigenvalues, then 2k ≤ n.
By Theorem 3.1 (iii), ifĠ has two eigenvalues, thenṠ has at most two eigenvalues. We consider them in more detail.

Theorem 3.3. Let a signed graphĠ with n vertices and two eigenvalues, λ and µ, be decomposed into the star complement kK 1 for µ and the signed graphṠ induced by the star set. Then either
(i)Ṡ is isomorphic to kK 1 and λ = −µ, or (ii)Ṡ has two eigenvalues: λ + µ with multiplicity k and µ with multiplicity n − 2k.
Proof. Let e i (1 ≤ i ≤ k) denote the ith vector of the canonical basis of R k . Observe that B x = 0 holds for every non-zero vector x ∈ R k , as otherwise we would have −λµIx = BB x = 0, which is impossible. By i.e., λ + µ is an eigenvalue of AṠ. As the kernel of B trivially intersects the space R k , the multiplicity of λ + µ is at least k. Denote by µ the average value of the n − 2k remaining eigenvalues (taken with their repetition). Considering tr(AĠ) = 0 and tr(AṠ) = 0, we get kλ+ (n − k)µ = 0 and k(λ+µ)+(n−2k)µ = 0, respectively. These equalities give (n−2k)(µ−µ) = 0. Hence, either 2k = n or µ = µ. In the former case, we immediately arrive at (i). In the latter, sinceṠ is a subgraph ofĠ, using the interlacing argument we get that µ is the unique remaining eigenvalue ofṠ with multiplicity n − 2k; hence, (ii).
Observe that in Theorem 3.3 (i),Ġ is bipartite and kK 1 is a star complement for both λ and µ. We proceed with a closer characterization. A matrix M is said to be orthogonal if M M = M M = rI, for r = 0. (There is an equivalent approach with r = 1.)

Corollary 3.4. Under the assumptions of Theorem 3.3,Ṡ is totally disconnected if and only if the matrix B of (2.1) is orthogonal.
Proof. IfṠ is totally disconnected, then by Theorem 3.3 (i), λ = −µ, which in conjunction with (3.3) gives Conversely, if B is orthogonal, then it follows by Theorem 3.1 (i) thatṠ is totally disconnected.
Let us take a break with an example illustrating the result of Theorem 3.3 (i). has two symmetric eigenvalues and admits a decomposition into two isomorphic totally disconnected signed graphs. Each of them is a star complement for both eigenvalues ofĠ.
A particular case of Theorem 3.3 (ii) is considered in the following subsection. Another particular case is treated in the forthcoming Subsection 3.2.
3.1. Case:Ġ has two eigenvalues, both ≥ −2. IfĠ is connected and its least eigenvalue is ≥ −2, thenĠ is either a signed line graph or an exceptional signed graph. An exceptional signed graph has a representation in the root system E 8 , but not in any D k . We presume that the reader is familiar with these and some related root systems (like E 7 or E 6 ), and also with the concept of their positive roots. In any case, all the necessary definitions can be found in [4,7]. The following two results, taken respectively from [9] and [7], are needed. (ii) the negative quadrangle or the signed multigraph obtained by inserting the (parallel) negative edge between the vertices of degree 2 of the path with four vertices, (iii) the complete graph K n , or (iv) a connected signed doubled regular graph with n vertices, in all cases, for n ≥ 3.
Obviously, a signed root graph K 1,n−1 gives a signed line graph that is switching equivalent to K n . Both signed root graphs of Theorem 3.6 (ii) give the negative quadrangle whose spectrum is 3), we compute the spectra of the signed line graphs of Theorem 3.6 (iii) & (iv); they are given by , where F is a regular graph with n vertices and degree r.

Theorem 3.7 ([7]
). IfĠ is an exceptional signed graph with n vertices, then n ≥ 6, the multiplicity of −2 in its spectrum is n − k, for some k ∈ {6, 7, 8}, andĠ is an induced subgraph of a signed graph represented by all the positive roots of E k . By [7], for 6 ≤ k ≤ 8, the signed graph represented by all the positive roots of E k is unique (up to the switching isomorphism). Denote it byṀ k . By the same reference (and this can also be computed directly), its spectrum is given bẏ We proceed with the following result concerning star complements in signed line graphs. (iii): A totally disconnected subgraph of L(K n ) is induced by a matching in K n . If such a graph is a star complement, then for −2 the matching must count n − 1 edges, and for n − 2 it must count n−1 2 edges. Both are impossible (under our assumption on n), as a matching in K n has at most n 2 edges. (iv): Similarly, a totally disconnected star complement in L(F ) is induced by a matching inF . For the eigenvalue −2, such a matching must count n edges, which occurs only if F has a perfect matching. For the eigenvalue 2(r − 1), the size of a matching inF must attain n(r − 1), which occurs only if r = 2 and F has a perfect matching, and we are done. The opposite implication is verified directly.
We continue with star complements in exceptional signed graphs. Theorem 3.9. If an exceptional signed graphĠ with two eigenvalues has a totally disconnected star complement, then its spectrum is For the first spectrum, i = 1 in the second and i = 2 in the third, the star complement corresponds to both eigenvalues. In the remaining cases, it corresponds to −2.
Proof. By Theorem 3.7, −2 is an eigenvalue of a signed graph whenever it has at least nine vertices. Considering those with fewer than nine vertices (they can be found in [4]), we arrive at the unique solution with spectrum Assume further that −2 is an eigenvalue ofĠ and denote the other eigenvalue by λ. Consider totally disconnected star complements for −2. By Theorem 3.7, the multiplicity of λ is k (6 ≤ k ≤ 8) andĠ is an induced subgraph ofṀ k . The corresponding star complement is kK 1 .
The case k = 6 cannot occur, as the size of the largest co-clique inṀ 6 is 4, and so the size of the largest co-clique inĠ is at most 4. In the remaining cases the size of the largest co-clique inṀ k is equal to k. Considering tr(AĠ) = 0 and tr(A 2Ġ ) = n (n being the number of vertices inĠ), we get n = k 2 (λ + 2). Since n is an integer and is bounded by the number of vertices inṀ k , we arrive at the desired spectra.
The remainder of the proof, concerning star complements for λ, follows by Theorem 3.3. Example 3.10. We confirm the existence ofĠ with the second spectrum of Theorem 3.9 for every i and for the third spectrum for every even i, by a successive removal of a selected co-clique. Precisely, we start withṀ k (k ∈ {7, 8}), then remove the roots that correspond to a selected co-clique of size k, and check whether the signed graph obtained contains a co-clique of the same size. Then we continue with this signed graph in the role ofṀ k .
3.2. Case: k = µ 2 . Here, by Theorem 3.1 (i), all the entries of B are ±1s. We consider the case whereĠ is decomposed into the star complement µ 2 K 1 and either a totally disconnected graph or a signed graph which is switching equivalent to a complete graph. Note that, by (3.2), b i , b j = 0 cannot occur if µ 2 is odd. In other words, forṠ totally disconnected, µ 2 must be even. Theorem 3.11. If a signed graphĠ with n vertices is decomposed into the star complement µ 2 K 1 (for µ) and a signed graphṠ, then n ≤ 2µ 2 and (i) ifṠ is totally disconnected, the spectrum ofĠ is µ n−µ 2 , 0 2µ 2 −n , (−µ) n−µ 2 ;