### Citations

7498 |
Matrix Analysis
- Horn, Johnson
- 1988
(Show Context)
Citation Context ...iii) and (iv). By Theorem 6, (iv) and (v) are equivalent. It remains to show the equivalence of (ii) and (iii). We recall the well-known identity (which is a special case of Sylvester’s identity, see =-=[9]-=-,p.22): If A is an n × n matrix, then for any i = j, det A(i|i) det A(j|j) − det A(i|j) det A(j|i) = (det A) det A(i, j|i, j). (27) 11In particular, in the present situation, A is a matrix over IF2,... |

1299 |
Introduction to graph theory
- West
- 2001
(Show Context)
Citation Context ...ds. block graph, adjacency matrix, determinant, tree, line graph AMS Subject Classifications. 15A15, 05C05. 11 Introduction We consider undirected graphs with no loops or parallel edges. We refer to =-=[11]-=- for graph theoretic preliminaries. Recall that a block in a graph is a maximal connected subgraph that has no cut-vertex ([11],p.15). The complete graph with n vertices is denoted by Kn. A block grap... |

13 |
An interrelation between line graphs, eigenvalues, and matroids
- Doob
- 1973
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Citation Context ...nce the summation in (2) is zero. Therefore the result is proved in this case. We therefore assume that none of the blocks among B2, . . . , Bq is pendant. Consider (−1) n−k ∑ (α1 − 1) · · · (αk − 1) =-=(5)-=- where the summation is over all k-tuples (α1, . . . , αk) of nonnegative integers satisfying the following conditions: k∑ (i) αi = n i=1 (ii) for any nonempty S ⊂ {1, . . . , k}, ∑ αi ≤ |V (GS)|. (6)... |

6 |
On nonsingular trees and a reciprocal eigenvalue property, Linear Mulitilinear Algebra 54(2006
- Barik, Neumann, et al.
(Show Context)
Citation Context ... that a tree is nonsingular over reals if and only if it has a perfect matching. Moreover, when a tree is nonsingular over reals, there is a formula for its inverse in terms of alternating paths, see =-=[3]-=-,[4], and the references contained therein. These two results motivated this work. Since a tree is a block graph, it is natural to investigate the adjacency matrix of a general block graph. We now des... |

3 | Tošić: More about singular line graphs of trees
- Marino, Sciriha, et al.
(Show Context)
Citation Context ...her singular or has zero as a simple eigenvalue. Thus the nullity of the line graph of a tree over the reals is at most one. An alternative proof of the result and some related results were proved in =-=[10]-=-. An easier proof of the same result and some extensions were recently proved in [2],[6]. It may be remarked that in view of Theorem 10, the nullity of the line graph of a tree is at most one over IF2... |

2 |
On the nullity of line graphs of trees. Discrete Math
- Sciriha, Gutman
- 2001
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Citation Context ... G \ {i} has at most two components, both of which must be nonsingular by Theorem 9. This contradiction shows that y = x and hence x is the unique null vector of A. It was shown by Gutman and Sciriha =-=[8]-=- that over the reals, the line graph of a tree is either singular or has zero as a simple eigenvalue. Thus the nullity of the line graph of a tree over the reals is at most one. An alternative proof o... |

1 |
Hindustan Book Agency
- Bapat, Graphs, et al.
- 2010
(Show Context)
Citation Context ...e adjacency matrix of a block graph over the field of reals as well as over the two-element field {0, 1}, which we denote as IF2. For basic definitions concerning matrices associated with graphs, see =-=[1]-=-. A graph is said to be nonsingular over reals if its adjacency matrix is nonsingular over reals. Similarly, a graph is said to be nonsingular over IF2 if its adjacency matrix is nonsingular over IF2.... |

1 |
On graphs with signed inverses, Networks
- Buckley, Doty, et al.
- 1988
(Show Context)
Citation Context ...t a tree is nonsingular over reals if and only if it has a perfect matching. Moreover, when a tree is nonsingular over reals, there is a formula for its inverse in terms of alternating paths, see [3],=-=[4]-=-, and the references contained therein. These two results motivated this work. Since a tree is a block graph, it is natural to investigate the adjacency matrix of a general block graph. We now describ... |

1 |
Spanning trees and line graph eigenvalues
- Ghorbani
- 2012
(Show Context)
Citation Context ...tree over the reals is at most one. An alternative proof of the result and some related results were proved in [10]. An easier proof of the same result and some extensions were recently proved in [2],=-=[6]-=-. It may be remarked that in view of Theorem 10, the nullity of the line graph of a tree is at most one over IF2 as well. This observation is a special case of a result obtained by Doob [5, Theorem 2.... |

1 |
Xinmao Wang and Yaokun Wu, Does the lit-only restriction make any difference for the σ-game and the σ +-game
- Goldwasser
- 2009
(Show Context)
Citation Context ...llity of the line graph of a tree is at most one over IF2 as well. This observation is a special case of a result obtained by Doob [5, Theorem 2.9] and can also be derived easily from some results in =-=[7]-=-. Also, the first part (n even) of Theorem 10 is contained in [7, Fact 22]. 5 Flower A block graph is called a flower if it has at most one cut vertex. By F (b1, . . . , bk) we denote a flower with k ... |