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Register a new userDistance matrices of a tree: two more invariants, and in a unified framework
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Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q
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In this paper we introduce the notion of $Sigma$-colouring of a graph $G$: For given subsets $Sigma(v)$ of neighbours of $v$, for every $vin V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that appear together in some $Sigma(v)$ receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of graphs embedded in a surface. We prove a general result for graphs embeddable in a fixed surface, which implies asymptot
It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a new characterization of the Shannon capacity $Theta$ via Strassens Positivstellensatz: $Theta(bar{G}) = inf_f f(G)$, where $f : mathsf{Graph} to mathbb{R}_+$ ranges over all monotone semiring homomorphisms. Constructing and classifying graph invariants $mathsf{Graph} to mathbb{R}_+$ which are monotone under graph homomorphisms, additive under join, and multiplicative under disjunctive product is therefore of major interest. We call such invariants semiring-homomorphic. The only known such invariants are all of a fractional nature: the fractional chromatic number, the projective rank, the fractional Haemers bounds, as well as the Lovasz number (with the latter two evaluated on the complementary graph). Here, we provide a unified construction of these invariants based on linear-like semiring families of graphs. Along the way, we also investigate the additional algebraic structure on the semiring of graphs corresponding to fractionalization. Linear-like semiring families of graphs are a new concept of combinatorial geometry different from matroids which may be of independent interest.
The emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices $i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices with edge weights are square matrix of same size. The distance $d_{ij}$ between the vertices $i$ and $j$ is the sum of the weight matrices of the edges in the unique path from $i$ to $j$. In this article we establish a characterization for the trees in terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we establish a necessary and sufficient condition for the distance matrix $D$, with matrix weights, to be invertible and the formula for the inverse of $D$, if it exists. Also we study some of the properties of the distance matrices of matrix weighted trees in connection with the Laplacian matrices, g-inverses and eigenvalues.
We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism invariant for unlabeled rooted trees. Then, we generalize the polynomial to unlabeled unrooted trees and we show that the generalized polynomial is a complete isomorphism invariant for unlabeled unrooted trees.
A rooted tree $T$ with vertex labels $t(v)$ and set-valued edge labels $lambda(e)$ defines maps $delta$ and $varepsilon$ on the pairs of leaves of $T$ by setting $delta(x,y)=q$ if the last common ancestor $text{lca}(x,y)$ of $x$ and $y$ is labeled $q$, and $min varepsilon(x,y)$ if $minlambda(e)$ for at least one edge $e$ along the path from $text{lca}(x,y)$ to $y$. We show that a pair of maps $(delta,varepsilon)$ derives from a tree $(T,t,lambda)$ if and only if there exists a common refinement of the (unique) least-resolved vertex labeled tree $(T_{delta},t_{delta})$ that explains $delta$ and the (unique) least resolved edge labeled tree $(T_{varepsilon},lambda_{varepsilon})$ that explains $varepsilon$ (provided both trees exist). This result remains true if certain combinations of labels at incident vertices and edges are forbidden.
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