On the maximal energy among orientations of a tree

Juan Rada, Juan Monsalve

Abstract


The trace norm of a digraph is the trace norm of its adjacency matrix, i.e.
the sum of its singular values. Given a bipartite graph $G$, it is well
known that the sink-source orientations have minimal trace norm among all
orientations of $G$. In this paper we show that the balanced orientations of
$G$ attain the maximal trace norm when $G$ is a tree with separated
branching vertices or when $G$ is a double-star tree. We give examples of
trees (with adjacent branching vertices) where non-balanced orientations
have maximal trace norm and raise the question in general: which
orientations of a tree have maximal trace norm?


Keywords


Trace norm; digraph; orientation of a tree; extremal values.

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References


bibitem[Agudelo & Rada(2016)]{agu-15} {bf Agudelo, N. & Rada, J. (2016).} Lower bounds of Nikiforov's energy

over digraphs, Linear Alg. Appl. {bf496} 156-164.

bibitem[Agudelo et al.(2016)]{agu-16} {bf Agudelo, N., de la Pe~{n}a, J.A. & Rada, J. (2016).} Extremal values

of the trace norm over oriented trees, Linear Algebra Appl. {bf505}

-268.

bibitem[Cvetkovi'{c} et al.(2010)]{cve-10} {bf Cvetkovi'{c}, D., Rowlinson, P. & Simi'{c}, S. (2010).} emph{An

Introduction to the Theory of Graph Spectra}. Cambridge University Press.

bibitem[Day & So(2007)]{day-07} {bf Day, J. & So, W. (2007).} Singular value inequality and graph energy

change, Electron. J. Linear Algebra {bf 16} 291-299.

bibitem[Fan(1951)]{fan-51} {bf Fan, K. (1951).} Maximum properties and inequalities for the

eigenvalues of completely continuous operators, Proc. Nat. Acad. Sci. USA,

{bf 37} 760-766.

bibitem[Gutman(1978)]{gut-78} {bf Gutman, I. (1978).} The energy of a graph. Ber. Math.-Statist. Sekt.

Forschungszentrum Graz {bf 103} 1-22.

bibitem[Horn & Johnson(2013)]{hor-13} {bf Horn, R. & Johnson, C. (2013).} emph{Matrix Analysis}, Cambridge

University Press.

bibitem[Kharaghani & Tayfeh-Rezaie(2008)]{kha-08} {bf Kharaghani, H. & Tayfeh-Rezaie, B. (2008).} On the energy of

(0,1)-matrices, Linear Alg. Appl. {bf 429}, 2046-2051.

bibitem[Li et al.(2012)]{li-12} {bf Li, X., Shi, Y. & Gutman, I. (2012).} emph{Graph energy},

Springer-Verlag, New York.

bibitem[Monsalve & Rada(2019)]{mon-18} {bf Monsalve, J. & Rada, J. (2019).} Oriented bipartite graphs with

minimal trace norm, Linear and Multilinear Algebra {bf 67} (6) 1121-1131.

bibitem[Monsalve et al.(2019)]{mon-19}{bf Monsalve, J., Rada, J. & Shi, Y. (2019).} Extremal values of energy over oriented bicyclic graphs, Appl. Math. Comput. {bf342} 26-34.

bibitem[Nikiforov(2007)]{nik-07} {bf Nikiforov, V. (2007).} The energy of graphs and matrices, J. Math.

Anal. Appl. {bf326} 1472-1475.

bibitem[So et al.(2010)]{so-10} {bf So, W., Robbiano, M., de Abreu, N. M. M. & Gutman, I. (2010).}

Applications of a theorem by Ky Fan in the theory of graph energy, Lin.

Algebra Appl. {bf432} 2163-2169.

bibitem[Mathematica, Version 10.0.(2014)]{wo-10} {bf Wolfram research, Inc. Mathematica, Version 10.0.}


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