Graph Invertibility

McLeman, Cam; McNicholas, Erin

doi:10.1007/s00373-013-1319-7

Cited by 17 publications

(21 citation statements)

References 8 publications

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“…So we have the following corollary. For signed graphs with a unique Sachs subgraph, the simply invertible property implies that the inverse weight function is integral (also call integral inverse) as described in the following theorem, which also generalizes and extends Theorem 2.1 in [19] for bipartite graphs with a unique perfect matching. Theorem 3.3.…”

Section: Signed Graphs With a Unique Sachs Subgraphmentioning

confidence: 67%

“…In the following, we consider two important families of graphs, one is called stellated graphs [13,26] and the other is called corona graphs [19]. Let G be a graph.…”

Section: Signed Graphs With a Unique Sachs Subgraphmentioning

confidence: 99%

“…[8]). Based on these two different definitions, the invertibility of bipartite graphs with unique perfect matching has been discussed in [7,14,19,25]. However, their methods can not be easily used for non-bipartite graphs.…”

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confidence: 99%

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Graph invertibility and median eigenvalues

Yang

Mandal

et al. 2017

Linear Algebra and its Applications

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Let (G, w) be a weighted graph with a weight-function w : E(G) → R\{0}. A weighted graph (G, w) is invertible to a new weighted graph if its adjacency matrix is invertible. A graph inverse has combinatorial interest and can be applied to bound median eigenvalues of a graph such as have physical meanings in Quatumn Chemistry. In this paper, we characterize the inverse of a weighted graph based on its Sachs subgraphs that are spanning subgraphs with only K2 or cycles (or loops) as components. The characterization can be used to find the inverse of a weighted graph based on its structures instead of its adjacency matrix. If a graph has its spectra split about the origin, i.e., half of eigenvalues are positive and half of them are negative, then its median eigenvalues can be bounded by estimating the largest and smallest eigenvalues of its inverse. We characterize graphs with a unique Sachs subgraph and prove that these graphs has their spectra split about the origin if they have a perfect matching. As applications, we show that the median eigenvalues of stellated graphs of trees and corona graphs belong to different halves of the interval [−1, 1].

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Section: Signed Graphs With a Unique Sachs Subgraphmentioning

confidence: 67%

“…In the following, we consider two important families of graphs, one is called stellated graphs [13,26] and the other is called corona graphs [19]. Let G be a graph.…”

Section: Signed Graphs With a Unique Sachs Subgraphmentioning

confidence: 99%

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Graph invertibility and median eigenvalues

Yang

Mandal

et al. 2017

Linear Algebra and its Applications

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“…The inverse of (G, w) is defined as a weighted graph (G −1 , w −1 ) whose vertex set is V (G −1 ) = V (G) and whose edge set is E(G −1 ) = {ij | (A −1 w ) ij = 0}, and whose weight function is w −1 (ij) = (A −1 w ) ij . Note that this definition of graph inverse is different from the definitions given in [5] and [12].…”

Section: Inverses Of Weighted Graphsmentioning

confidence: 98%

Inverses of Bipartite Graphs

Yang¹,

Ye²

2017

Combinatorica

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Let G be a bipartite graph and its adjacency matrix A. If G has a unique perfect matching, then A has an inverse A −1 which is a symmetric integral matrix, and hence the adjacency matrix of a multigraph. The inverses of bipartite graphs with unique perfect matchings have a strong connection to Möbius functions of posets. In this note, we characterize all bipartite graphs with a unique perfect matching whose adjacency matrices have inverses diagonally similar to non-negative matrices, which settles an open problem of Godsil on inverses of bipartite graphs in [Godsil, Inverses of Trees, Combinatorica 5 (1985) 33-39].

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“…This way of introducing invertibility has the appealing property that inverting an inverse graph gives back the original graph. For a survey of results and other approaches to graph inverses we recommend [11].…”

Section: Introductionmentioning

confidence: 99%

On a construction of integrally invertible graphs and their spectral properties

Pavlíková¹,

Ševčovič

2017

Linear Algebra and its Applications

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Godsil (1985) defined a graph to be invertible if it has a non-singular adjacency matrix whose inverse is diagonally similar to a nonnegative integral matrix; the graph defined by the last matrix is then the inverse of the original graph. In this paper we call such graphs positively invertible and introduce a new concept of a negatively invertible graph by replacing the adjective 'nonnegative' by 'nonpositive in Godsil's definition; the graph defined by the negative of the resulting matrix is then the negative inverse of the original graph. We propose new constructions of integrally invertible graphs (those with non-singular adjacency matrix whose inverse is integral) based on an operation of 'bridging' a pair of integrally invertible graphs over subsets of their vertices, with sufficient conditions for their positive and negative invertibility. We also analyze spectral properties of graphs arising from bridging and derive lower bounds for their least positive eigenvalue. As an illustration we present a census of graphs with a unique 1-factor on m ≤ 6 vertices and determine their positive and negative invertibility.

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Graph Invertibility

Cited by 17 publications

References 8 publications

Graph invertibility and median eigenvalues

Graph invertibility and median eigenvalues

Inverses of Bipartite Graphs

On a construction of integrally invertible graphs and their spectral properties

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