Proof of a Conjecture of Graham and Lovasz concerning Unimodality of Coefficients of the Distance Characteristic Polynomial of a Tree

Abstract

We establish a conjecture of Graham and Lovász that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal and prove they are log-concave. We also establish upper and lower bounds on the location of the peak.

Spectra of Variants of Distance Matrices of Graphs and Digraphs: A Survey

“…, d n−2 (T ) is at most 2 3 n and is at least n−2 1+d . It was also shown in [2] that the sequence d 0 (H ), . .…”

“…Bounds on the location of the peak for trees were presented (that paper also includes a more refined upper bound than the one listed next that depends on the structure of the tree). Theorem 2.12 [2] Let T be a tree on n ≥ 3 vertices with diameter d. The peak location of the normalized coefficients d 0 (T ), d 1 (T ), . .…”

“…1 The following graphs are determined by their D spectrum [17,26,30,39]: Theorem 7. 2 The following graphs are determined by their D Q spectrum [6,7,16]:…”

“…The directed strongly regular graph = (8, 4, 3, 1, 3), as shown in Fig. 10, has spectrum spec D ( ) = {10, −2 (5) , 0 (2) }. Applying Proposition 8.10, has order 8 and spec D ( ) = {10 (8 −1 ), −2(8 −1 )…”

Constructions in combinatorics via neural networks

“…In [1], the coefficients d k = 2 k 2 n−2 |δ k | are called the normalized coefficients of the characteristic polynomial of the distance matrix of T . Graham and Lovász [25] conjectured that for any tree T , the sequence of normalized coefficients d 0 (T ), .…”