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The maximum number of Parter vertices of acyclic matrices
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Orientador(es)
Resumo(s)
A vertex v of the underlying graph of a symmetric matrix A is called ‘Parter’ if the nullity of the matrix obtained from A by removing the row and column indexed by v is more than the nullity of A. Let A be a singular symmetric matrix with rank r whose underlying graph is a tree. It is known that the number of Parter vertices of A is at most r−1. We prove that when r is odd this number is at most r−2. We characterize the trees where these bounds are achieved.
Descrição
The authors would like to thank anonymous referees for their helpful comments and suggestions which considerably improved the presentation of the paper. The authors except the third one were supported by Portuguese Funds through the Portuguese Foundation for Science and Technology (FCT). The first, second, and fifth authors were supported by FCT (Fundacao para a Ciencia e Tecnologia), under the project UIDB/04721/2020 (CEAFEL - Centro de Analise Funcional e Estruturas Lineares). The fourth author was supported within the scope of the project UIDB/00297/2020 (CMA - Centro de Matematica e Aplicacoes). This work was started in August 2017 while the third author was visiting CEAFEL at the Faculty of Sciences in the University of Lisbon. He is sincerely grateful to CEAFEL for the hospitality and support in August 2017 and March 2018. He also wishes to express his gratitude for the travel grant he received from the Institute for Research in Fundamental Sciences (IPM) in August 2017.
Palavras-chave
Acyclic matrix Nullity Parter vertex Tree Theoretical Computer Science Discrete Mathematics and Combinatorics