The average distance of a vertex $v$ connected graph $G$ is the arithmetic mean distances from to all other vertices $G$. proximity $\pi(G)$ and remoteness $\rho(G)$ are minimum maximum $G$, respectively. In this paper, we give upper bounds on for graphs given order, degree degree. Our sharp apart an additive constant.

With any surjective rational map \(f: \mathbb {P}^n \dashrightarrow {P}^n\) of the projective space, we associate a numerical invariant (ML degree) and compute it in terms naturally defined vector bundle \(E_f \longrightarrow {P}^n\).

Given a simple graph $G$, denote by $\Delta(G)$, $\delta(G)$, and $\chi'(G)$ the maximum degree, minimum chromatic index of respectively. We say $G$ is $\Delta$-critical if $\chi'(G)=\Delta(G)+1$ $\chi'(H)\le \Delta(G)$ for every proper subgraph $H$ overfull $|E(G)|>\Delta(G) \lfloor |V(G)|/2 \rfloor$. Since matching in can have size at most $\lfloor \rfloor$, it follows that $\chi'(G) = \Delta...

A k-colouring of a graph G is a mapping c from the set of vertices of G to the set {1, . . . , k} of colours such that adjacent vertices receive distinct colours. Such a k-colouring is called acyclic, if for every two distinct colours i and j, the subgraph induced by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, every cycle in G has at l...

We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood (ML) degree counts number critical points function restricted model. prove that ML generic is determined by its Newton polytopes and equals mixed volume related Lagrange equations.