Invertibility of Graphs with a Unique Perfect Matching

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INVERTIBILITY OF GRAPHS WITH A UNIQUE PERFECT MATCHING

PAVLÍKOVÁ Soňa (SK), and ŠEVČOVIČ Daniel (SK)

Abstract. In this paper we investigate invertibility of graphs with a unique perfect matching, i.e. graphs having a unique 1-factor. We recall the new notion of the so-called negatively invertible graphs investigated by the authors in the recent paper. It is an extension of the classical deﬁnition of an inverse graph due to Godsil. We characterize all graphs with a unique perfect matching on m ≤ 6 vertices with respect to their positive and negative invertibility. We show that negatively invertible graphs exhibit properties like selﬁnvertibility which cannot be observed within the class of positively invertible non-bipartite graphs with a unique perfect matching.

Keywords and phrases: Integrally invertible graph, positively an negatively invertible graphs, 1-factor, unique perfect matching. Mathematics subject classiﬁcation: Primary 05C50; Secondary 15A09

1 Introduction and preliminaries arXiv:1612.02234v1 [math.CO] 7 Dec 2016 The main purpose of this paper is to investigate invertibility of graphs with a unique perfect matching (graphs having a unique 1-factor) (c.f. Pavlíková et al. [10, 9]). Following our recent paper [12], we recall the new notion of the so-called negatively invertible graphs. It is an extension of the classical deﬁnition of an inverse graph due to Godsil [4]. Our goal is to characterize all graphs with a unique perfect matching on m ≤ 6 vertices with respect to their positive and negative invertibility. According to the result due to Sachs (c.f. [2, p. 32]), it is known that, for a bipartite graph G having a unique perfect matching, the adjacency matrix A is integrally invertible. The aim of this paper is to investigate a larger class of graphs which are 6

1 23456 1 2345 6

6 6 5

1 2345 1 2345 1 234

4 3 6 4 3 6 6 4 3

1 2 5 1 2 5 1 2 5

4 3 6

1 2 5

2 3 2 3 2 3 4 6 4 6 4 6

1 5 1 5 1 5

Fig. 1. A graph representing the structural graph of a chemical molecule of fulvene not necessarily bipartite but still have a unique 1-factor and an integrally invertible adjacency matrix. 4 5 6 Let G = (V,E) be an undirected graph with the set of vertices V and the set of edges E. By AG we denote its adjacency matrix. Conversely, if A is a {0, 1}-symmetric matrix then GA denotes the graph with1 the23 adjacency matrix A. The spectrum σ(G) of an undirected graph G consists of eigenvalues of its adjacency symmetric matrix AG, i.e. σ(G) = σ(AG) = {λ, λ is an eigenvalue of AG} (c.f. [3, 2]). If the spectrum does not contain zero then the adjacency matrix A is invertible. −1 Clearly, a graph GA has the integral inverse matrix A if and only if the determinant det(A) = ±1 (c.f. Kirkland and Akbari [6]). The inverse matrix A−1 however may contain positive as well as negative entries. In fact, it is well known that the inverse matrix of the adjacency matrix of a graph is nonnegative if and only if the graph is a union of isolated edges (c.f. Harary [5]). Nevertheless, the concept of an inverse graph is indeed based on the inverse of its adjacency matrix. In [4] Godsil deﬁned a graph to be invertible if the inverse of its (non-singular) adjacency matrix is diagonally similar (c.f. [13]) to a non-negative matrix.

Deﬁnition 1 (Godsil [4]) An undirected vertex-labeled graph GA is called positively invertible (or invertible in Godsil’s sense) if inverse of its adjacency matrix is signable to a nonnegative integer matrix, i.e. there exists a diagonal matrix DA containing ±1 elements and such that all elements of the matrix DAA−1DA are nonnegative integers −1 A −1 A only. The inverse graph H = GA is deﬁned by the adjacency matrix AH = D A D .

Unfortunately, the class of positively invertible graphs does not contain many important examples of graphs having integral inverse matrix. Among them there is a graph representing the chemical organic molecule of fulvene (see Fig. 1). The inverse of its adjacency matrix is integral but it is not signable to a nonnegative matrix. The structural graph of fulvene however plays an important role in computational chemistry as it represents a chemical molecule. For instance, spectral properties of inverse graphs are important in order to compute the so-called binding energies of molecular orbitals. Motivated by this shortcoming of positive invertibility of a graph, in the recent paper [12], Pavlíková and Ševčovič introduced the new concept of the so-called negative invertibility of a graph with integral inverse of its adjacency matrix.

Deﬁnition 2 (Pavlíková and Ševčovič [12]) An undirected vertex-labeled graph GA is called negative invertible if the inverse matrix of its adjacency matrix is signable to a nonpositive integer matrix, i.e. there exists a diagonal matrix DA containing ±1 elements and such that all elements of the matrix DAA−1DA are nonpositive integers only. −1 A −1 A The inverse graph H = GA is deﬁned by the adjacency matrix AH = −D A D .

The adjacency matrix A of the fulvene graph and its inverse matrix A−1 are as follows: 0 1 0 0 1 0  0 0 0 0 1 −1 1 0 1 0 0 0  0 0 1 0 0 −1     0 1 0 1 0 0 −1  0 1 0 0 −1 1  A =   ,A =   . 0 0 1 0 1 1  0 0 0 0 0 1      1 0 0 1 0 0  1 0 −1 0 0 1  0 0 0 1 0 0 −1 −1 1 1 1 −2

The matrix A−1 is signable to a nonpositive matrix by the diagonal matrix DA = diag(1, 1, −1, −1, −1, 1). Both positive as well as negatively invertible graphs have the following useful property. A −1 A If G = GA is positively invertible then AH = D A D is the adjacency matrix of −1 A −1 A −1 −1 the inverse graph H = G . As A = D AH D we have the identity G = (G ) . This property also remains true in the case of negatively invertible graphs. Indeed, if A −1 A G = GA is negatively invertible then AH = −D A D is the adjacency matrix of the −1 A −1 A −1 −1 inverse graph H = G . As A = −D AH D we again obtain G = (G ) . In the next section we recall the complete list of graphs with a unique perfect matching on m ≤ 6 vertices discovered by Pavlíková and Ševčovič [12]. In addition, we present their inverse graphs and maximal subgraphs with a unique perfect matching. Inter- estingly enough, there are no positively invertible non-bipartite graphs with a unique perfect matching that are selﬁnvertible. On the other hand, there exist two negatively invertible graphs which are selﬁnvertible.

2 Connected graphs with a unique 1-factor In this section we present the complete list of invertible graphs on m ≤ 6 vertices with a unique 1-factor. Recall that a graph G has a unique 1-factor if there exists a unique subgraph spanning G and such that each vertex has the degree 1. Notice that any graph having a 1-factor should have even number of vertices.

For m = 2 the graph K2 is the unique connected graph and it has a unique 1-factor. It is a positively and negatively invertible bipartite graph.

For m = 4 there are two connected graphs Q1,Q2 with a unique 1-factor shown in Fig. 2. The graph Q1 is bipartite and selﬁnvertible. The graph Q2 is positively invertible but it is not selﬁnvertible. Next we analyse the connected graphs on m = 6 vertices with a unique 1-factor. We have the following theorem.

Theorem 1 [12] There exist 20 undirected connected graphs on m = 6 vertices with a unique 1-factor shown in Figures 3,4,5,6,7. All of them have invertible adjacency matrices. Except of the graph H19 they are integrally invertible. 4

1 234 1 23

3 44

2 143 2 14 1 234 1 23 1 234 1 23 −1 −1 Q1 (Q1) Q1 = (Q1) 3 3 2 144 3 2 14

2 143 2 14 1 234 1 23 −1 −1 Q2 (Q2) Q2 ⊆ (Q2) 3 Fig. 2. Invertible graphs on four vertices with a unique perfect 2 143 matching,2 14 their inverse graphs and maximal subgraphs with a unique perfect matching

There are three bipartite graphs H1,H2, and H6 which are simultaneously positively and negatively invertible. There are twelve graphs

H3,H4,H7,H8,H9,H13,H14,H15,H16,H17,H18,H20,

which are positively invertible. The three graphs H5,H10, and H12 are negatively invertible. The integrally invertible graph H11 is neither positively nor negatively invertible. The graphs H8, and H18 are iso-spectral but not isomorphic.

Recall that the proof of this result is based on the well-known Kotzig’s theorem. It states that a graph with a unique 1-factor should contain a bridge belonging to the 1-factor sub-graph. Using this characterization one can construct all 20 connected graphs on 6 vertices having a unique 1-factor. We refer the reader to [12] for details.

2.1 Positively and negatively invertible (bipartite) graphs with a unique perfect matching on m = 6 vertices

Recall that a graph GB is called bipartite if the set of vertices can be divided into two disjoint subsets such that every edge connects a vertex from the ﬁrst subset into the one from the second subset.

Theorem 2 (Pavlíková and Ševčovič [12]) Let G be an integrally invertible graph. Then G is a bipartite graph if and only if G is simultaneously positively and negatively invertible.

In [8] McLeman and McNicholas considered graphs which can be obtained from a given connected graph by attaching pendant edges to each of vertices. They proved that such graphs are selﬁnvertible, i.e. H−1 = H. An example illustrating their result is the graph H2 which is obtained from a path with three vertices by adding pendant edges to each of them. Tab. 1. The family of graphs on 6 vertices with a unique 1-factor and their signability. Source: [12]

Graph invertibility

H1 positively and negatively invertible (bipartite) H2 positively and negatively invertible (bipartite) H3 positively invertible H4 positively invertible H5 negatively invertible H6 positively and negatively invertible (bipartite) H7 positively invertible H8 positively invertible H9 positively invertible H10 negatively invertible H11 noninvertible with integral inverse of the adjacency matrix H12 negatively invertible 3 H13 positively invertible H14 positively2 5 4 invertible1 6 3 2 1 6 5 4 5 H15 positively invertible 6 H16 positively1 invertible 1 6 H17 positively invertible 1 23456 1 2345 6452 H18 positively3 invertible 3 6254 2 143 6 H19 noninvertible with nonintegral inverse of the adjacency matrix 6 6 5 H20 positively invertible

1 2345 1 2345 1 234

1 2 4 6 1 2 4 3 3 6 4 3 6 6 4 3

1 23456 1 233 6 455 1 2 6 5 5 4 1 2 5 1 2 5 −1 6 −1 H1 (H1) H6 = (H1) 3 5 2 4 3 6 6 6 5 6 3 1 6 4 1 2 5 1 2345 1 2345 1 234 1 1 2345 23456 2 5 4 1 6 3 2 1 6 5 4 6 −1 5 −1 4 3 H62 4 3(H2)6 6 4H2 3=3 (H2) 3 3 2 3 2 2 5 1 2 6 1 6 5 1 6 4 6 6 4 4 6 6 4 4 6 3 1 1 3 6452 3 6254 2 1 143 1 1 2345 1 232 5 45 1 1 232 54 1 525 51 5 3 2 5 2 5 4 1 6 3 2 1 6 5 4 5 4 3 6 44 33 6 6 6 4 3 −1 −1 H6 1 (H6) H1 = (H6) 6 4 6 5 136 4 6 3 1 2 5 1 Fig.2 3.5 Positively and1 negatively2 5 invertible (bipartite) graphs with a 1 2 1 5 3 643 1522 3 6254 2 143 unique4 perfect2 matching, their inverses and maximal4 subgraphs5 6 with 4 3 6 a unique perfect1 32 matching 4 5 2 1 52

2 3 36 5 62 5 3 4 12 233 1 2 5 In Fig. 3 we show4 all6 three bipartite graphs4 6 on m = 6 vertices with4 a6 unique perfect 3 5 2 matching. According to the4 previous1 2 theorem they3 1 are simultaneously2 621 positive as well 4 6 3 1 5 1 5 1 5 2 3 2 3 2 3 4 6 1 6 4 46 4 6 3 6 5 6 5 4 5 3 4 1 5 2 1 1 5 1 5 3 5 2 5 3 2 5 1 6 4 6 4

4 5 6 1 6 4 5 1 3 2

4 5 6 1 23 3 2 5 1 6 4 6 4 4 6 5 136 4 6 3 1 23 5 1 3 2

1 32 4 5 2 1 52

4 6 5 136 4 6 3

621 4 6 3 1 32 4 5 2 1 52

5 3 4 1 5 2

621 4 6 3

5 3 4 1 5 2 as negative invertible.

2.2 Positively invertible graphs with a unique perfect matching on m = 6 vertices In this section we present the complete list of twelve non-bipartite graphs with a unique perfect matching on m = 6 vertices which are positively invariant (see Fig. 4,5). In- terestingly, none of them is selﬁnvertible and their inverse graphs contain vertices with −1 −1 −1 −1 loops with even multiplicity. The inverse graphs (H4) , (H8) , (H13) , (H15) contain multiple edges. There are four graphs denoted by H9,H13,H14, and H18 such that −1 they are maximal subgraphs of their inverse graphs, i.e. Hi ⊆ (Hi) , i = 9, 13, 14, 18. −1 −1 There are two pairs of graphs (Hi,Hj) such that Hi ⊆ (Hj) and Hj ⊆ (Hi) for pairs (i, j) ∈ {(3, 7), (15, 17)}.

2.3 Negatively invertible graphs with a unique perfect matching on m = 6 vertices In this section we present the complete list of all three non-bipartite graphs with a unique perfect matching on m = 6 vertices which are negatively invariant. The graphs H5, and H12 are selfinvertible. Recall that according to the results due to McLeman and McNicholas [8] a graph which is obtained from a given connected graph by attaching pendant edges to each of vertices is selfinvertible. Among negatively invertible graphs with unique perfect matching on m = 6 vertices as an example illustrating this property one can present the graph H5. In contrast to positively invertible bipartite graphs there exists the graph H12 which is selfinvertible but it cannot be obtained by means of the construction due to McLeman and McNicholas.

2.4 Noninvertible graphs with a unique perfect matching on m = 6 vertices In this section we present two remaining graphs on m = 6 vertices which are not invertible. The ﬁrst graph is denoted by H11 and it has the adjacency matrix which is integrally invertible. Nevertheless, A−1 is neither positively nor negatively signable, so H11 is not invertible. The graph H19 has the adjacency matrix with det(A) = 3 and, −1 as a consequence, A is not integer matrix. Hence H19 is noninvertible as well.

3 Conclusions In this note we investigated invertible graphs with a unique perfect matching on m ≤ 6 vertices. We recalled classical concept of graph inversion due to Godsil which we referred to as positive invertibility of a graph. We also recalled the new concept of negative invertibility proposed by the authors in [12]. By inspecting properties of the inverse graphs we showed that negatively invertible graphs exhibits properties like selﬁnvertibility which cannot be observed within the class of positively invertible non- bipartite graphs with a unique perfect matching on m ≤ 6 vertices.

Acknowledgements The research was supported by the APVV Research Grant 0136-12 (SP) and VEGA grant 1/0780/15 (DŠ). 6 3 1 23456 1 2345 2 5 4 1 6 3 2 1 6 5 4 5 6

1 6 1 6 6 5

6452 3 6254 3 1 2345 1 22314453 1 234

4 3 6 4 3 6 6 4 3

4 1 2 1 2 3 5 1 2 1 2 5 1 2 5

4 3 6 3 6 5 6 5 4

3 3 5 2 1 2 5 6 2 5 4 1 6 3 2 1 6 5 4 5 1 6 4 1 23456 1 2345 1 2 3 1 2 3 6 2 3 4 6 4 6 6 4 6 3 2 1 3 6452 3 6254 52 143 6 6 6 4 6 64 5 1 5 1 5 1 5 5 1 3 2 1 23451 6 231 2345451 2345 1 234 6 4 3 6 4 3 6 6 4 3 4 1 2 3 1 2 6 4 6 5 136 6 4 65 3

4 5 6 4 3 6 4 5 6 1 233 6 6 5451 21 6 5235 4 451 21 5234 1 2 5 1 32 4 5 2 1 52 −1 3−1 3 H53 2 (H3) H7 ⊆ (H3) 1 23456 41 233 6 4514 2334 63 6 16 24 53 1 23 2 5 6 4 1 6 3 2 1 6 5 4 1 5 1 6 4 4 623 4643 6 3 4636 6 1 2 5 6 1 21 525 5 1 2 5 1 3 1 6 1 3 23456 1 2345 5 32 4 1 5 2 1 2345 14 233 6 456 11 3 2236 5 4 1 52 15 1 2 5 2 3 5 644 1 526 3 2 31 626 5 544 2 6 2 3 6 4 2 3 6 4 2 5 3143 2 5 4 1 6 −13 2 1 6 5 4 −1 H4 (4H46) H17 ⊆ (4H4)6 4 6 4 3 6 4 1 3 236 456 14 233 45 6 5 6 5 6 15 1 3 2 1 6 6 1 2 5 6 1 1 5 1 5 1 6 1 5 6 3 64526 1 3 23625 4554 21 14233 45 1 234 1 23451 26 15 23451 2 5 1 2 5 3 3 2 4 66 3 5 64252136 3 622544 63 3 2 143 4 1 2 3 1 2 3 6 1 234 6 45 1 234 456 4 13 236 4 4 6 4 6 4 3 4 3 6 6 6 5 −1 −1 H7 1 32 (H7) 4 3 5 2 H3 ⊆1 (H527) 1 4 53 6 41 5 6 6 4 1 3 5 3 6 5 1 6 5 4 1 1 1 2345 1 2345 1 2 5 2 5 2 5 1 2 5 234 44 1 5 2 6 3 1 2 1 31 5 2 1 2 5 2 5 4 3 26 5 4 3 6 4 3 6 6 4 3 3 1 2 4 1 2 621 4 6 3 1 3 6 5 6 5 4 2 3 2 4 33 6 232 3 4 6 4 6 1 6 44 6 1 2 5 1 2 5 1 2 5 3 5 2 1 2 5 4 5 6 3 6 5 5 3 4 6 5 4 1 5 2 1 −1 −1 1 5 1 52 H5 8 1(H8)5 H9 ⊆ (H8) 4 3 6 2 3 5 2 3 5 1 1 6 4 3 3 3 1 23 2 6 4 2 6 4 2 5 2 3 4 2 32 2 43 6 4 6 4 6 1 2 5 4 6 4 6 5 1 4 6 3 2 1 6 4 3 2 5 1 1 5 1 5 1 5 6 4 6 4 1 5 1 1 6 53 1 5 3 3 3 2 4 5 2 6 2 4 6 4 6 4 6 3 2 5 1 31 2 4 −61 5 136−1 5 4 6 3 H9 (H9) H9 ⊆6 (H4 9) 6 4 1 1 1 5 1 235 5 1 32 5 1 4 5 2 3 2 1 52 4 6 5 136 4 6 3 4 5 6 4 5 6

1 32 4 5 2 1 52 1 23 4 6 1361 23 4 6 3 5 621 4 6 3 4 5 6 −1 −1 H13 (H13) H13 ⊆ (H13)

1 4 5 2 1 52 Fig. 4. Positively invertible32 graphs with a unique62 perfect51 3 matching,4 4 61 35 2 1 23 2 6 1 their inverses and maximal subgraphs with a unique perfect

matching (1. part) 5 3 1 5 5 43 4 2 621 4 6 3 156 References 5 3 4 1 5 2 [1] R. B. Bapat, and E. Ghorbani, Inverses of triangular matrices4 3 and2 bipartite graphs, Linear Algebra and its Applications 447 (2014), 68-73. [2] D. Cvetković, M. Doob, and H. Sachs, Spectra of graphs - Theory and application, Deutscher Verlag der Wissenchaften, Berlin, 1980; Academic Press, New York, 1980. 3

2 5 4 1 6 3 2 1 6 5 4 6 5

1 1 66 1 23456 1 2345 6 3 64521 233 6245546 1 22314453 6 6 5 6

1 2345 1 23456 1 234 6 5

4 3 6 4 3 6 6 4 3 1 2345 1 2345 1 234 6 4 1 2 3 1 2

1 2 5 1 2 5 4 3 6 1 2 5 4 3 6 6 4 3 1 23456 1 2345 3 6 5 6 6 5 4 4 3 6 1 6 1 1 6 6 3 5 2 5 2 5 2 5 2 5 1 1 2 5 4 1 5 232 456 2345 1 2345 1 2345 1 234 4 3 6 6 1 6 4 6 4 3 6 4 3 6 6 4 6 3 6 5 2 3 1 6 2 3 3 2 3 1 2 5 1 23456 1 2345 4 6 4 6 4 6 1 2345 3 1 22345 15 231 4 6 1 2 5 1 2 5 1 2 5 6 4 6 4 1 1 1 5 5 5 6 6 5 4 3 6 2 3 4 3 6 62 4 3 3 2 3 3 4 3 6 5 4 1 6 3 42 6 4 6 3 1 2345 1 2345 1 234 6 1 2 5 1 2 5 1 2 5 2 5 4 1 6 3 2 1 6 5 4 1 5 1 5 1 5 1 2 5 5 3 4 3 6 3 2 5 4 1 6 3 2 1 6 5 4 4 6 6 4 4 4 6 3 56 135 6 4 6 3 1 23456 11 4235 6 45 4 3 16 6 4 5 6 1 3 1 6 2 3 2 2 3 6 1 2 5 1 2 5 1 2 5 1 4 5 2 4 6 3 64524 6 13 32622 5 4546 42 5142 3 1 52 6 1 23 1 2 5 1 23 3 64526 3 62545 2 143 2 6 1 4 3 6 1 5 1 5 1 5 −1 −1 H14 4 1 5(6H146 ) H144 ⊆ 3(H146) 4 5 6 4 3 2 463 3 2 463 3 3 2 3 1 2345 1 2345 1 234 4 6 4 6 4 6 1 2 5 5 43 621 4 6 3 4 3 6 4 6 3 1 6 2 5 61 145 23 5 1 231 15 2 5 1 12 5 5 1 23 4 1 2 3 1 2 4 1 2 3 1 2 156 2 3 2 3 2 3 45 33 4 461 3 5 2 463 4 5 6 4 3 6 4 5 6 4 6 4 6 4 6 1 2 5 1 2 5 1 2 5 3 3 66 55 66 55 4 4 4 3 2 1 5 1 5 1 5 4 1 3 236 1 2 5 1 23 6 1 2 5 1 2 5 1 2 5 3 3 55 22 4 5 6 4 3 6 4 5 6 H (H )−1 H ⊆ (H )−1 4 3 463 463 15 15 17 15 1 2 5 1 1 2 5 1 23 1 1 66 44 23 2 6 1 6 1 2 5 1 2 5 1 2 5 4 3 463 4 465 3 6 3 2 3 3 2 3 5 1 2 3 2 2 55 431 6 4 6 4 4 6 4 6 6 4 4 6 6 4 6 1 2 5 1 2 5 1 1232 5 156 5 1 3 2 −1 −1 1 5 1 5 5 1 1 H516 3 (H216) H13 ⊆ (H16)

4 3 2 136 4 6 64 6 5 3 4 6 5 136 4 6 3

1 23456 1 231 3245 4 5 2 1 52 4 5 4 3 6 4 5 6 6 1 32 4 5 2 1 52 6 −1 −1 H17 (H17) H15 ⊆ (H17) 6 6 5 1 1 1 23 2 5 621 23 4 6 3

1 2345 1 2345 1 23621 4 4 6 3 4 3 463 463 5 3 4 1 5 2 4 3 6 4 3 6 6 4 3 6 5 3 4 3 1 5 2 6 1 2 5 1 2 5 1 2 5 1 −1 1 −451 1 2 5 1 2 5 2 5 1 4 2 1 5 6 H318 2 1 236 (5H18454) 6 H18 ⊆23(H18) 5 6 4 3 6 1 4 3 6 1 6 3 6 6 6 5

3 6452 3 6254 2 143 1 2 5 1 2 45 1 2 5 1 2345 1 2345 1 234 −1 −1 H20 (H20) H4 ⊆ (H20) 4 3 6 4 3 6 6 4 3 2 3 2 3 Fig.2 5.3 Positively invertible graphs with a unique perfect matching, 4 6 4 6 6 their4 inverses and maximal subgraphs with a unique perfect 4 1 2 3 1 1 2 2 5 1 2 5 1 2 5 1 5 1 5 1 5 matching (2. part) 4 3 6 3 6 5 6 5 4 [3] D. Cvetković, I. Gutman, and S. Simić, On selfpseudo-inverse graphs, Univ. Beograd3 5 Publ.2 Elektrotehn. Fak. Ser. Mat. ﬁz., No. 602 – 633, (1978), 111–117. 1 2 5 [4] C. D. Godsil, Inverses of Trees, Combinatorica 5 (1985) 1, 33–39. 4 5 6 4 3 6 4 5 6 [5] F.1 Harary,6 4 and H. Minc, Which nonnegative matrices are self-inverse? Math. Mag. 3 (Math. Assoc. of America)2 493 (1976) 2, 91–92. 2 2 3 4 6 4 6 4 6 1 23 1 2 5 1 23 3 2 5 1 1 65 4 1 6 5 4 1 5 4 3 463 463

5 1 3 2

6 1 2 5 1 2 5 1 2 5

4 6 5 136 4 6 3 4 5 6

1 32 4 5 2 1 52 1 23

621 4 6 3

5 3 4 1 5 2 3

2 5 4 1 6 3 2 1 6 5 4 5 6 6 1 1 6

1 23451 6 231 2345456 1 2345 3 6452 3 6254 2 143 3 6 6

6 6 6 5 6 5 2 5 4 1 6 3 2 1 6 5 4 5 1 23451 231 2345451 2345 6 1 234 1 234 1 1 6 6 4 1 2 3 1 2 4 3 3 6 4 43 63 6 46 43 63 6 4 3 1 23456 1 2345 3 6452 3 6254 2 143 1 23456 1 2345 6 2 5 4 1 6 3 2 1 6 5 4 3 6 6 6 5 1 25 5 1 21 5 52 4 5 1 21 52 5 1 2 5 6 6 5 6 1 6 5 1 6 5 3 4 32 6 4 3 6 1 2345 1 2345 1 234 1 2345 1 2345 1 234 3 6452 3 6254 2 143 4 1 2 3 1 2 4 3 4 6 3 6 4 43 36 6 61 1464 343 1 −1 −1 6 2H5 5 2(H55) H5 = (H5)

3 6 5 6 5 4 1 2 1 5 2 5 1 12 25 5 1 1 2 2 55 2 2 3 2 3 23 3 2 5 23 1 3 2 3 1 3 1 42 6 46 644 6 4 6 64 4 6 4 6 4 6 5 4 2 4 3 6 3 3 2 1 5 1 5 15 1 5 1 3 15 2 5 1 5 1 2 5 3 6 5 6 5 4 6 1 1 6 4 −1 −1 2 5 H10 (H10) H11 ⊆ (H10) 3 5 2 3 3 1 23456 1 2345 2 2 4 2 6 3 5 136 4 6 3 2 2 3 4 6 2 3 4 6 3 2 3 4 6 5 1 6 4 6 1 6 4 4 6 6 4 6 6 4 4 4 1 5 1 5 1 55 66 5 6 6 5 1 32 4 5 2 1 52 1 5 1 5 5 11 5 3 2 3 2 5 1 1 2345 1 1 236 4 1 23−6 1 4 2345 1 −1234 H12 (H12) H12 = (H12)

5 41 3 6 3 1 2 4 3 6 6 4 3 4 6 5 Fig.13 6. Negatively6 invertible624 graphs6 3 with a unique4 perfect6 3 matching, 4 5 6 their inverses and maximal subgraphs with a unique perfect 4 5 6 1 2 5 matching3 41 2 1 5 2 1 2 1 23 1 324 6 5 4 135 62 5 1 4526 3 5 5

1 23 4 3 6 1 32 4 5 2 1 523 6

621 4 6 3 1 2 5 1 2 45

621 H 4 6 3 H 5 3 4 11 1 5 2 19 2 3 2 3 2 3 Fig. 7. Noninvertible graphs with a unique perfect matching: H11 5 3 4 4 6 1 5 2 4 6 6 has integral inverse of its adjacency matrix but it is neither 4

positively nor1 negatively5 invertible,1 H195 is not integrally1 invertible5

[6] S. J. Kirkland, and S. Akbari, On unimodular graphs, Linear Algebra and its Applications 421 (2007), 3–15. [7] S. J. Kirkland, and R. M. Tifenbach, Directed intervals and the dual of a graph, Linear Algebra and4 its5 Applications6 4314 (2009),3 6 792–807. 4 5 6 [8] C. McLeman, and E. McNicholas, Graph Invertibility, Graphs and Combinatorics (2014) 30, 977-10021 23 1 2 5 1 23

[9] S. Pavlíková, 4Graphs3 with a463 unique 1-factor46 and3 matrices, PhD Thesis (in Slovak), Comenius University, Bratislava, 1994, www.iam.fmph.uniba.sk/studium/phd/pavlikova 6 1 2 5 1 2 5 1 2 5 [10] S. Pavlíková, and J. Krč-Jediný, On the inverse and dual index of a tree, Linear and Multilinear Algebra 28 (1990), 93–109. [11] S. Pavlíková, A note on inverses of labeled graphs. to appear: Australasian Journal on Combinatorics 2016. [12] S. Pavlíková, and D.Ševčovič, On a Construction of Integrally Invertible Graphs and their Spectral Properties, submitted, 2016. [13] T. Zaslavsky, Signed graphs, Discrete Applied Math. 4 (1982), 47–74.

Current address Pavlíková Soňa, RNDr., CSc. Institute of Information Engineering, Automation, and Mathematics, FCFT, Slovak Technical University, 812 37 Bratislava, Slovak republic [email protected] Ševčovič Daniel, Prof., RNDr., CSc. Institute of Mathematics and Physics, Faculty of Mechanical Engineering, Slovak Uni- versity of Technology in Bratislava, Nám. slobody 17, 812 31 Bratislava, Slovak republic [email protected]