Let G be a graph of order n and let μ be an eigenvalue of multiplicity m. A star complement for μ in G is an induced subgraph of G of order n −m with no eigenvalue μ. In this paper, we study maximal and regular graphs which have Kr,s + tK1 as a star complement for 1 as the second largest eigenvalue. It turns out that some well known strongly regular graphs are uniquely determined by such a star...

*Correspondence: [email protected] 1School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, Jiangsu 224002, P.R. China Full list of author information is available at the end of the article Abstract A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013) and Liu et al. (Electron. J. Linear ...

An eigenvalue of a graph G is called main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Hoffman [A.J. Hoffman, On the polynomial of a graph, Amer. Math. Monthly 70 (1963) 30–36] proved that G is a connected k-regular graph if and only if n ∏t i=2(A− λiI ) = ∏t i=2(k − λi) · J , where I is the unit matrix and J the all-one matrix and λ1 = k, λ2, . . . , λt ar...

A spectral inverse problem concerns the reconstruction of parameters a parent graph from prescribed data subgraphs. Also referred to as P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, aim is seek sets determine uniquely. Other related problems, including Polynomial Problem (PRP), involve recovery invariants. The PRP seeks extract spectrum deck cards each showing vertex-deleted...

For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on graph only through degree and vertex-Cheeger constant. We also prove analogous result a large class irregular graphs, obtained as spanning subgraphs graphs. Using Babai, obtain operator in terms diameter degree.

Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as synchronization and cascading processes on networks. Here we develop a theory for estimating the change of the largest eigenvalue of the adjacency matrix or the extreme ...

Our first aim in this note is to prove some inequalities relating the eigenvalues of a Hermitian matrix with the eigenvalues of its principal matrices induced by a partition of the index set. One of these inequalities extends an inequality proved by Hoffman in [9]. Secondly, we apply our inequalities to estimate the eigenvalues of the adjacency matrix of a graph, and prove, in particular, that ...

The purpose of this paper is to develop a “calculus” on graphs that allows graph theory to have new connections to analysis. For example, our framework gives rise to many new partial differential equations on graphs, most notably a new (Laplacian based) wave equation (see [FTa, FTc]); this wave equation gives rise to a partial improvement on the Chung-Faber-Manteuffel diameter/eigenvalue bound ...