| Summary: | 博士 === 國立交通大學 === 應用數學系所 === 106 === The spectral radius of a square matrix C is the largest magnitude of an eigenvalue of C and the spectral radius of a graph G is the spectral radius of the adjacency matrix of G. Let G be a bipartite graph with e edges without isolated vertices. It was known that the spectral radius of G is at most the square root of e, and the upper bound is attained if and only if G is a complete bipartite graph. Our first result extends this result to find the maximum spectral radius of a non-complete bipartite graph with e edges under the assumption that (e - 1; e + 1) is not a pair of twin primes.
Bhattacharya, Friedland and Peled conjectured that a non-complete bipartite graph which has the maximum spectral radius with given e and bi-order (p, q) is obtained from a complete bipartite graph by deleting edges incident
to a common vertex. We find counter examples of this conjecture. Under the additional assumption e>=pq-q or under the assumption p<=5, where p<=q, we prove a weaker version of the above conjecture that drops the non-complete
assumption of the bipartite graph.
To handle the problem above, we study the spectral radius of a nonnegative matrix C which takes the square of the adjacency matrix of G as a special case. For a general nonnegative matrix C, we give a new approach to obtain lower bounds and upper bounds of the spectral radius of C which are the spectral radii of matrices obtained by suitably reweighting the entries in a row of C keeping the row-sum unchanged. This method helps us to find many spectral bounds of C easily.
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