On the spectral moments of trees with a given bipartition

For two given positive integers $p$ and $q$ with $p\leqslant q$, we denote \mathscr{T}_n^{p, q}={T: T is a tree of order $n$ with a $(p, q)$-bipartition}. For a graph $G$ with $n$ vertices, let $A(G)$ be its adjacency matrix with eigenvalues $\lambda_1(G), \lambda_2(G), ..., \lambda_n(G)$ in non-increasing order. The number $S_k(G):=\sum_{i=1}^{n}\lambda_i^k(G)\,(k=0, 1, ..., n-1)$ is called the $k$th spectral moment of $G$. Let $S(G)=(S_0(G), S_1(G),..., S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, one has $G_1\prec_s G_2$ if for some $k\in {1,2,...,n-1}$, $S_i(G_1)=S_i(G_2) (i=0,1,...,k-1)$ and $S_k(G_1)<S_k(G_2)$ holds. In this paper, the last four trees, in the $S$-order, among $\mathscr{T}_n^{p, q} (4\leqslant p\leqslant q)$ are characterized.Comment: 11 pages, 7 figure

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum