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Let $D$ be a strongly connected digraph. The average distance $bar{sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $rho(D)$ and proximity $pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $pi(D)$ and $rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $pi(T)=rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $pi(D)=rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $pi(D)=rho(D).$ We describe such a family for undirected graphs as well.
In this paper, we study independent domination in directed graphs, which was recently introduced by Cary, Cary, and Prabhu. We provide a short, algorithmic proof that all directed acyclic graphs contain an independent dominating set. Using linear alg
Let $G$ be a directed graph such that the in-degree of any vertex $G$ is at least one. Let also ${mathcal{tau}}: V(G)rightarrow Bbb{N}$ be an assignment of thresholds to the vertices of $G$. A subset $M$ of vertices of $G$ is called a dynamic monopol
Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length $N$ i
Given a directed graph, an equivalence relation on the graph vertex set is said to be balanced if, for every two vertices in the same equivalence class, the number of directed edges from vertices of each equivalence class directed to each of the two
Let $G$ be a simple $n$-vertex graph and $c$ be a colouring of $E(G)$ with $n$ colours, where each colour class has size at least $2$. We prove that $(G,c)$ contains a rainbow cycle of length at most $lceil frac{n}{2} rceil$, which is best possible.