Extending a conjecture of Graham and Lov\'{a}sz on the distance characteristic polynomial

TL;DR

This paper extends Graham and Lovász's conjecture on the unimodality of the normalized coefficients of the distance characteristic polynomial from trees to block graphs, proving unimodality and identifying peaks in specific cases.

Contribution

It proves the unimodality of the normalized coefficients for block graphs and determines the peak for certain extremal uniform block graphs with small diameter.

Findings

Proved unimodality of the normalized coefficients for block graphs.

Identified the peak position for extremal uniform block graphs with small diameter.

Extended the conjecture from trees to a broader class of graphs.

Abstract

Graham and Lov\'{a}sz conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order $n$ is unimodal with the maximum value occurring at $\lfloor\frac{n}{2}\rfloor$. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter.