Bounds for eigenvalues of a graph

2022

The smallest possible number of distinct eigenvalues of a graph G, denoted by q(G), has a combinatorial bound in terms of unique shortest paths in the graph. In particular, q(G) is bounded below by k, where k is the number of vertices of a unique shortest path joining any pair of vertices in G. Thus, if n is the number of vertices of G, then n − q(G) is bounded above by the size of the complement (with respect to the vertex set of G) of the vertex set of the longest unique shortest path joining any pair of vertices of G. The purpose of this paper is to commence the study of the minor-monotone floor of n − k, which is the minimum of n − k among all graphs of which G is a minor. Accordingly, we prove some results about this minor-monotone floor.

Linear Algebra and its Applications, 2003

Let G = (V , E) be a simple graph on vertex set V = {v 1 , v 2 ,. .. , v n }. Further let d i be the degree of v i and N i be the set of neighbors of v i. It is shown that max d i + d j − |N i ∩ N j | : 1 i < j n, v i v j ∈ E is an upper bound for the largest eigenvalue of the Laplacian matrix of G, where |N i ∩ N j | denotes the number of common neighbors between v i and v j. For any G, this bound does not exceed the order of G. Further using the concept of common neighbors another upper bound for the largest eigenvalue of the Laplacian matrix of a graph has been obtained as max 2 d 2 i + d i m i : 1 i n , where m i = j d j − |N i ∩ N j | : v i v j ∈ E d i .

Linear Algebra and its Applications, 1992

We give lower bounds for the smallest eigenvalue of the Laplacian of corresponding undirected connected multigraphs in terms of the expansion coeffkients and norm estimates. Upper bounds for the convergence rate of certain nonnegative irreducible symmetric matrices and irreducible diagonally symmetrizable stochastic matrices are given.

arXiv (Cornell University), 2022

In this paper we show that the d-dimensional algebraic connectivity of an arbitrary graph G is bounded above by its 1-dimensional algebraic connectivity, i.e., a d (G) ≤ a 1 (G), where a 1 (G) corresponds the well-studied second smallest eigenvalue of the graph Laplacian.

Journal of Inequalities and Applications, 2014

Let G be a simple connected graph of order n, where n ≥ 2. Its normalized Laplacian eigenvalues are 0 = λ 1 ≤ λ 2 ≤ · · · ≤ λ n ≤ 2. In this paper, some new upper and lower bounds on λ n are obtained, respectively. Moreover, connected graphs with λ 2 = 1 (or λ n-1 = 1) are also characterized.

Gazi University Journal of Science, 2015

Let ‫ܩ‬ be a simple connected graph and its Laplacian eigenvalues be ߤ ଵ ߤ ଶ ⋯ ߤ ିଵ ߤ ൌ 0. In this paper, we present an upper bound for the algebraic connectivity ߤ ିଵ of ‫ܩ‬ and a lower bound for the largest eigenvalue ߤ ଵ of ‫ܩ‬ in terms of the degree sequence ݀ ଵ , ݀ ଶ ,. .. , ݀ of ‫ܩ‬ and the number |ܰ ∩ ܰ | of common vertices of ݅ and ݆ ሺ1 ݅ ൏ ݆ ݊ሻ and hence we improve bounds of Maden and Büyükköse [14].

Computational and Applied Mathematics

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 -1λ 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 -1λ 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.

Linear Algebra and its Applications, 2007

This paper is a survey of the second smallest eigenvalue of the Laplacian of a graph G, best-known as the algebraic connectivity of G, denoted a(G). Emphasis is given on classifications of bounds to algebraic connectivity as a function of other graph invariants, as well as the applications of Fiedler vectors (eigenvectors related to a(G)) on trees, on hard problems in graphs and also on the combinatorial optimization problems. Besides, limit points to a(G) and characterizations of extremal graphs to a(G) are described, especially those for which the algebraic connectivity is equal to the vertex connectivity. N.M.M. de Abreu / Linear Algebra and its Applications 423 (2007) Among all eigenvalues of the Laplacian of a graph, one of the most popular is the second smallest, called by Fiedler , the algebraic connectivity of a graph. Its importance is due to the fact that it is a good parameter to measure, to a certain extent, how well a graph is connected. For example, it is well-known that a graph is connected if and only if its algebraic connectivity is different from zero.

Discrete Mathematics, 2004

The eigenvalues of a graph are the eigenvalues of its adjacency matrix. This paper presents some upper and lower bounds on the greatest eigenvalue and a lower bound on the smallest eigenvalue.

arXiv: Spectral Theory, 2019

Let $R$ be a Hermitian matrix. The energy of $R$, $\mathcal{E}(R)$, corresponds to the sum of the absolute values of its eigenvalues. In this work it is obtained two lower bounds for $\mathcal{E}(R).$ The first one generalizes a lower bound obtained by Mc Clellands for the energy of graphs in $1971$ to the case of Hermitian matrices and graphs with a given nullity. The second one generalizes a lower bound obtained by K. Das, S. A. Mojallal and I. Gutman in 2013 to symmetric non-negative matrices and graphs with a given nullity. The equality cases are discussed. These lower bounds are obtained for graphs with $m$ edges and some examples are provided showing that, some obtained bounds are incomparable with the known lower bound for the energy $2\sqrt{m}$. Another family of lower bounds are obtained from an increasing sequence of lower bounds for the spectral radius of a graph. The bounds are stated for singular and non-singular graphs.