Geodesic
An interlacing theorem for matrices whose graph is a given tree
Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 3, pp. 245-254
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Let $A$ and $B$ be $(n\times n)$-matrices. For an index set $S\subset\{1,\ldots,n\}$, denote by $A(S)$ the principal submatrix that lies in the rows and columns indexed by $S$. Denote by $S'$ the complement of $S$ and define $\eta(A,B)=\sum\limits_S\det A(S)\det B(S')$, where the summation is over all subsets of $\{1,\ldots,n\}$ and, by convention, $\det A(\varnothing)=\det B(\varnothing)=1$. C. R. Johnson conjectured that if $A$ and $B$ are Hermitian and $A$ is positive semidefinite, then the polynomial $\eta(\lambda A,-B)$ has only real roots. G. Rublein and R. B. Bapat proved that this is true for $n\leq3$. Bapat also proved this result for any $n$ with the condition that both $A$ and $B$ are tridiagonal. In this paper, we generalize some little-known results concerning the characteristic polynomials and adjacency matrices of trees to matrices whose graph is a given tree and prove the conjecture for any $n$ under the additional assumption that both $A$ and $B$ are matrices whose graph is a tree. 
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C. da Fonseca. An interlacing theorem for matrices whose graph is a given tree. Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 3, pp. 245-254. http://geodesic.mathdoc.fr/item/FPM_2004_10_3_a12/

[1] Bapat R. B., “An interlacing theorem for tridiagonal matrices”, Linear Algebra Appl., 150 (1991), 331–340 | DOI | MR | Zbl

[2] Cvetković D. M., Doob M., Sachs H., Spectra of Graphs, Academic Press, New York, 1980 | MR | Zbl

[3] Ky Fan, Pall G., “Imbedding conditions for Hermitian and normal matrices”, Canad. J. Math., 9 (1957), 298–304 | DOI | MR | Zbl

[4] Godsil G., “Spectra of trees”, Ann. Discrete Math., 20 (1984), 151–159 | MR | Zbl

[5] Godsil G., Algebraic Combinatorics, Chapman and Hall, New York, 1993 | MR | Zbl

[6] Horn R. A., Johnson C. R., Matrix Analysis, Cambridge University Press, 1985 | MR | Zbl

[7] Johnson C. R., “The permanent-on-top conjecture: a status report”, Current Trends in Matrix Theory, eds. F. Uhlig, R. Grone, Elsevier Science, 1987, 167–174 | MR

[8] Johnson C. R., “A characteristic polynomial for matrix pairs”, Linear and Multilinear Algebra, 25 (1989), 289–290 | DOI

[9] Lovász L., Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979 | MR | Zbl

[10] Marshall A. W., Olkin I., Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, 1976 | MR | Zbl

[11] Rublein G., “On a conjecture of C. Johnson”, Linear and Multilinear Algebra, 25 (1989), 257–267 | DOI | MR | Zbl