Some new aspects of main eigenvalues of graphs (original) (raw)
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On main eigenvalues of certain graphs
2016
An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of harmonic graphs is analyzed and they are characterized in terms of their main eigenvalues without any restriction on its combinatorial structure. We give a necessary and sufficient condition for a graph G to have −1 − λ min as an eigenvalue of its complement, where λ min denotes the least eigenvalue of G. Also, we prove that among connected bipartite graphs, K r,r is the unique graph for which the index of the complement is equal to −1 − λ min. Finally, we characterize all paths and all double stars (trees with diameter three) for which the smallest eigenvalue is non-main. Main eigenvalues of paths and double stars are identified.
Some results on graphs with exactly two main eigenvalues
Applied Mathematics Letters, 2012
An eigenvalue of a graph G is called main if there is an associated eigenvector not orthogonal to j, the vector with each entry equal to 1. In this work, an error in a prior paper [Y. Hou and F. Tian, Unicyclic graphs with exactly two main eigenvalues, Appl. Math. Letters, 19 (2006), 1143-1147] is pointed out and the properties of the graphs with exactly two main eigenvalues and with pendent vertices are discussed. As an application, we obtain, together with known results, all connected bicyclic and tricyclic graphs with exactly two main eigenvalues.
A note on graphs with exactly two main eigenvalues
Linear Algebra and its Applications
In this note, we consider connected graphs with exactly two main eigenvalues. We will give several constructions for them, and as a consequence we show a family of those graphs with an unbounded number of distinct valencies. 2010 Mathematics Subject Classification. 05C50.
Recent progress on graphs with fixed smallest eigenvalue
arXiv: Combinatorics, 2020
We give a survey on graphs with fixed smallest eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory related to Hoffman graphs and the applications of Hoffman graphs to graphs with fixed smallest eigenvalue and large minimal valency; (ii) recent results on distance-regular graphs and co-edge regular graphs with fixed smallest eigenvalue and the characterizations of certain families of distance-regular graphs. At the end of the survey, we also discuss signed graphs with fixed smallest eigenvalue and present some new findings.
Eigenvector Centrality and Uniform Dominant Eigenvalue of Graph Components
ArXiv, 2021
Eigenvector centrality is one of the outstanding measures of central tendency in graph theory. In this paper we consider the problem of calculating eigenvector centrality of graph partitioned into components and how this partitioning can be used. Two cases are considered; first where the a single component in the graph has the dominant eigenvalue, secondly when there are at least two components that share the dominant eigenvalue for the graph. In the first case we implement and compare the method to the usual approach (power method) for calculating eigenvector centrality while in the second case with shared dominant eigenvalues we show some theoretical and numerical results.
On the distribution of eigenvalues of a simple undirected graph
Linear Algebra and its Applications, 1999
For a simple, undirected graph q n , let k i q n be the ith largest eigenvalue of q n. This paper presents mainly the following: 1. For n P 4, if q n is incomplete, then À n 2 T k n q n `À 1 1 4 nÀ3 nÀ1 q 2 X 2. Seven sucient and necessary conditions such that k 2 q n À1. 3. k 3 q n À1 implies that k j q n À1Y j 3Y 4Y F F F Y n À 1.
Extremal graph characterization from the bounds of the spectral radius of weighted graphs
Applied Mathematics and Computation, 2011
We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain a lower bound and an upper bound on the spectral radius of the adjacency matrix of weighted graphs and characterize graphs for which the bounds are attained.
Linear and Multilinear Algebra
We study regular graphs whose distance-2 graph or distance-1-or-2 graph is strongly regular. We provide a characterization of such graphs Γ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance d from every vertex, where d + 1 is the number of different eigenvalues of Γ. This can be seen as another version of the so-called spectral excess theorem, which characterizes in a similar way those regular graphs that are distance-regular.
Regular graphs with small second largest eigenvalue
Applicable Analysis and Discrete Mathematics, 2013
We consider regular graphs with small second largest eigenvalue (denoted by ?2). In particular, we determine all triangle-free regular graphs with ?2 ? ?2, all bipartite regular graphs with ?2 ? ?3, and all bipartite regular graphs of degree 3 with ?2 ? 2.
Electronic Journal of Linear Algebra, 2015
Given a simple graph G, let A(G) be its adjacency matrix. A principal submatrix of A(G) of order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known that the multiplicity of any eigenvalue of A(G) and such a principal submatrix can differ by at most one. Therefore, a vertex v of G is a downer vertex (neutral vertex, or Parter vertex) with respect to a fixed eigenvalue μ if the multiplicity of μ in A(G)âv goes down by one (resp., remains the same, or goes up by one). In this paper, we consider the problems of characterizing these three types of vertices under various constraints imposed on graphs being considered, on vertices being chosen and on eigenvalues being observed. By assigning weights to edges of graphs, we generalizeour results to weighted graphs, or equivalently to symmetric matrices.