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Connected resolvability of graphs

Varaporn Saenpholphat, Ping Zhang (2003)

Czechoslovak Mathematical Journal

For an ordered set W = { w 1 , w 2 , , w k } of vertices and a vertex v in a connected graph G , the representation of v with respect to W is the k -vector r ( v | W ) = ( d ( v , w 1 ) , d ( v , w 2 ) , , d ( v , w k ) ) , where d ( x , y ) represents the distance between the vertices x and y . The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W . A resolving set for G containing a minimum number of vertices is a basis for G . The dimension dim ( G ) is the number of vertices in a basis for G . A resolving set W of G is connected if the subgraph...

Coxeter polynomials of Salem trees

Charalampos A. Evripidou (2015)

Colloquium Mathematicae

We compute the Coxeter polynomial of a family of Salem trees, and also the limit of the spectral radii of their Coxeter transformations as the number of their vertices tends to infinity. We also prove that if z is a root of multiplicities m , . . . , m k for the Coxeter polynomials of the trees , . . . , k respectively, then z is a root for the Coxeter polynomial of their join, of multiplicity at least m i n m - m , . . . , m - m k where m = m + + m k .

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