On the nullity of graphs with pendent vertices

Li, Shuchao

doi:10.1016/j.laa.2008.04.037

Cited by 41 publications

(29 citation statements)

References 9 publications

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“…More results on the nullity (or rank) of graphs can be found in the book [28] and the paper [31]. For the nullity of some special classes of graphs, one can refer to the papers [10,12,[14][15][16]20,[23][24][25]27,32,34].…”

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confidence: 99%

A note on the nullity of unicyclic signed graphs

Fan

Wang

2013

Linear Algebra and its Applications

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confidence: 99%

A note on the nullity of unicyclic signed graphs

Fan

Wang

2013

Linear Algebra and its Applications

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“…connected graphs) with pendent vertices and nullity η (0 < η ≤ n). As corollaries of this characterization, some results in [9] can be obtained immediately. Moreover, all bipartite graphs (resp.…”

Section: Introduction Let G Be a Simple Undirected Graph With Vertexmentioning

confidence: 75%

On the characterization of graphs with pendent vertices and given nullity

Liu¹,

Huang²,

Chen³

2009

ELA

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Abstract. Let G be a graph with n vertices. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we characterize the graphs (resp. bipartite graphs) with pendent vertices and nullity η, where 0 < η ≤ n. Moreover, the minimum (resp. maximum) number of edges for all (connected) graphs with pendent vertices and nullity η are determined, and the extremal graphs are characterized.

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“…Observe that cospectral graphs have equal orders and sizes, and therefore if a connected graph is cospectral with a double comet, then such a graph must be a tree. Next, recall from [6] that the multiplicity of zero (also known as nullity) in the spectrum of a tree is n − 4 if and only if that tree is a generalized double comet C * (k 1 , k 2 , l), for 2 ≤ l ≤ 3. Similarly, the trees of nullity n−6 are those illustrated in Fig.…”

Section: Spectral Gap Of Treesmentioning

confidence: 99%

Trees with small spectral gap

2017

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Continuing the previous research, we consider trees with given number of vertices and minimal spectral gap. Using the computer search, we conjecture that this spectral invariant is minimized for double comet trees. The conjecture is confirmed for trees with at most 20 vertices; simultaneously no counterexamples are encountered. We provide theoretical results concerning double comets and putative trees that minimize the spectral gap. We also compare the spectral gap of regular graphs and paths. Finally, a sequence of inequalities that involve the same invariant is obtained.

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On the nullity of graphs with pendent vertices

Cited by 41 publications

References 9 publications

A note on the nullity of unicyclic signed graphs

A note on the nullity of unicyclic signed graphs

On the characterization of graphs with pendent vertices and given nullity

Trees with small spectral gap

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