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Your search keyword '"Adjacency matrix"' showing total 2 results
Start Over Descriptor "Adjacency matrix" Topic 05c50 Publication Year Range Last 10 years Journal discrete mathematics
2 results on '"Adjacency matrix"'

1. Conjectured bounds for the sum of squares of positive eigenvalues of a graph.

Author

Elphick, Clive, Farber, Miriam, Goldberg, Felix, and Wocjan, Pawel

Subjects
*SUM of squares, *EIGENVALUES, *GRAPHIC methods, *MATHEMATICAL bounds, *LOGICAL prediction
Abstract

A well known upper bound for the spectral radius of a graph, due to Hong, is that μ 1 2 ≤ 2 m − n + 1 if δ ≥ 1 . It is conjectured that for connected graphs n − 1 ≤ s + ≤ 2 m − n + 1 , where s + denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete q -partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general. [ABSTRACT FROM AUTHOR]

Published
2016
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2. Minimum supports of functions on the Hamming graphs with spectral constraints

Author

Alexandr Valyuzhenich and Konstantin V. Vorob'ev

Subjects
Discrete mathematics, Direct sum, 020206 networking & telecommunications, 0102 computer and information sciences, 02 engineering and technology, Characterization (mathematics), 01 natural sciences, Theoretical Computer Science, Combinatorics, Hamming graph, 010201 computation theory & mathematics, FOS: Mathematics, 0202 electrical engineering, electronic engineering, information engineering, Mathematics - Combinatorics, Discrete Mathematics and Combinatorics, Cardinality (SQL statements), Combinatorics (math.CO), Adjacency matrix, 05C50, Eigenvalues and eigenvectors, Mathematics
Abstract

We study functions defined on the vertices of the Hamming graphs $H(n,q)$. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\leq j\leq n$. For the case $n\geq i+j$ and $q\geq 3$ we find the minimum cardinality of the support of such functions and obtain a characterization of functions with the minimum cardinality of the support. In the case $n \frac{n}{2}$,$\,q\ge 5$., Comment: 17 pages, 3 figures

Published
2019

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