647 research outputs found

    Merging the A- and Q-spectral theories

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    Let GG be a graph with adjacency matrix A(G)A\left( G\right) , and let D(G)D\left( G\right) be the diagonal matrix of the degrees of G.G. The signless Laplacian Q(G)Q\left( G\right) of GG is defined as Q(G):=A(G)+D(G)Q\left( G\right) :=A\left( G\right) +D\left( G\right) . Cvetkovi\'{c} called the study of the adjacency matrix the AA% \textit{-spectral theory}, and the study of the signless Laplacian--the QQ\textit{-spectral theory}. During the years many similarities and differences between these two theories have been established. To track the gradual change of A(G)A\left( G\right) into Q(G)Q\left( G\right) in this paper it is suggested to study the convex linear combinations Aα(G)A_{\alpha }\left( G\right) of A(G)A\left( G\right) and D(G)D\left( G\right) defined by Aα(G):=αD(G)+(1−α)A(G),   0≤α≤1. A_{\alpha}\left( G\right) :=\alpha D\left( G\right) +\left( 1-\alpha\right) A\left( G\right) \text{, \ \ }0\leq\alpha\leq1. This study sheds new light on A(G)A\left( G\right) and Q(G)Q\left( G\right) , and yields some surprises, in particular, a novel spectral Tur\'{a}n theorem. A number of challenging open problems are discussed.Comment: 26 page

    On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions

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    Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains νn\nu_n of the nn-th eigenfunction satisfies n≥νnn\ge \nu_n. Here, we provide a new interpretation for the Courant nodal deficiency dn=n−νnd_n = n-\nu_n in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the nn-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.Comment: 22 pages, 6 figure

    Eigenvalue bounds for the signless laplacian

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    We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures

    Beyond graph energy: norms of graphs and matrices

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    In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia

    A theory of spectral partitions of metric graphs

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    We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012), 815--838] as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic -- rather than numerical -- results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in [Conti \textit{et al}, Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\ Henri Poincar\'e Anal.\ Non Lin\'eaire \textbf{26} (2009), 101--138], but we can also generalise some of them and answer (the graph counterparts of) a few open questions
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