UID:
almafu_9959238843502883
Format:
1 online resource (xi, 364 pages) :
,
digital, PDF file(s).
ISBN:
1-316-08660-7
,
1-107-36079-X
,
0-511-80151-3
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1-107-36837-5
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1-107-36570-8
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1-299-40898-2
,
1-107-36324-1
Series Statement:
London Mathematical Society student texts ; 75
Content:
This introductory text explores the theory of graph spectra: a topic with applications across a wide range of subjects, including computer science, quantum chemistry and electrical engineering. The spectra examined here are those of the adjacency matrix, the Seidel matrix, the Laplacian, the normalized Laplacian and the signless Laplacian of a finite simple graph. The underlying theme of the book is the relation between the eigenvalues and structure of a graph. Designed as an introductory text for graduate students, or anyone using the theory of graph spectra, this self-contained treatment assumes only a little knowledge of graph theory and linear algebra. The authors include many developments in the field which arise as a result of rapidly expanding interest in the area. Exercises, spectral data and proofs of required results are also provided. The end-of-chapter notes serve as a practical guide to the extensive bibliography of over 500 items.
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Title; Copyright; Contents; Preface; 1 Introduction; 1.1 Graph spectra; 1.2 Some more graph-theoretic notions; 1.3 Some results from linear algebra; Exercises; Notes; 2 Graph operations and modifications; 2.1 Complement, union and join of graphs; 2.2 Coalescence and related graph compositions; 2.3 General reduction procedures; 2.4 Line graphs and related operations; 2.5 Cartesian type operations; 2.6 Spectra of graphs of particular types; Exercises; Notes; 3 Spectrum and structure; 3.1 Counting certain subgraphs; 3.2 Regularity and bipartiteness; 3.3 Connectedness and metric invariants
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3.4 Line graphs and related graphs3.5 More on regular graphs; 3.5.1 The second largest eigenvalue; 3.5.2 The eigenvalue with second largest modulus; 3.5.3 Miscellaneous results; 3.6 Strongly regular graphs; 3.7 Distance-regular graphs; 3.8 Automorphisms and eigenspaces; 3.9 Equitable partitions, divisors and main eigenvalues; 3.10 Spectral bounds for graph invariants; 3.11 Constraints on individual eigenvalues; 3.11.1 The largest eigenvalue; 3.11.2 The second largest eigenvalue; Exercises; Notes; 4 Characterizations by spectra; 4.1 Spectral characterizations of certain classes of graphs
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4.1.1 Elementary spectral characterizations4.1.2 Graphs with least eigenvalue -2; 4.1.3 Characterizations according to type; 4.2 Cospectral graphs and the graph isomorphism problem; 4.2.1 Examples of cospectral graphs; 4.2.2 Constructions of cospectral graphs; 4.2.3 Statistics of cospectral graphs; 4.2.4 A comparison of various graph invariants; 4.3 Characterizations by eigenvalues and angles; 4.3.1 Cospectral graphs with the same angles; 4.3.2 Constructing trees; 4.3.3 Some characterization theorems; Exercises; Notes; 5 Structure and one eigenvalue; 5.1 Star complements
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7.5.3 Isoperimetric problems7.6 Expansion; 7.7 The normalized Laplacian matrix; 7.8 The signless Laplacian; 7.8.1 Basic properties of Q-spectra; 7.8.2 Q-eigenvalues and graph structure; 7.8.3 The largest Q-eigenvalue; Exercises; Notes; 8 Some additional results; 8.1 More on graph eigenvalues; 8.1.1 Graph perturbations; 8.1.2 Bounds on the index; 8.2 Eigenvectors and structure; 8.3 Reconstructing the characteristic polynomial; 8.4 Integral graphs; Exercises; Notes; 9 Applications; 9.1 Physics; 9.1.1 Vibration of a membrane; 9.1.2 The dimer problem; 9.2 Chemistry
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9.2.1 The Hückel molecular orbital theory
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English
Additional Edition:
ISBN 0-521-11839-5
Additional Edition:
ISBN 0-521-13408-0
Language:
English
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