For two vertices $u$ and $v$ in a connected graph $G$, the set $I(u,v)$ consists of all those vertices lying on a $u-v$ geodesic in $G$. For a set $S$ of vertices of $G$, the union of all sets $I(u,v)$ for $u,v\in S$ is denoted by $I\left(S\right)$. A set $S$ is convex if $I\left(S\right)=S$. The convexity number $\mathrm{c}on\left(G\right)$ is the maximum cardinality of a proper convex set in $G$. A convex set $S$ is maximum if $\left|S\right|=\mathrm{c}on\left(G\right)$. The cardinality of a maximum convex set in a graph $G$ is the convexity number of $G$. For a nontrivial connected graph $H$, a connected graph $G$ is an $H$-convex graph if $G$ contains...

In this paper the following results are proved: 1. Let $P_n$ be a path with $n$ vertices, where $n\ge 5$ and $n\ne 7,8$. Let $M$ be a matching in $P_n$. Then ${\left(P_n\right)}^4-M$ is hamiltonian-connected. 2. Let $G$ be a connected graph of order $p\ge 5$, and let $M$ be a matching in $G$. Then $G^5-M$ is hamiltonian-connected.

For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^k$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^k$ if the directed distance ${}^{\to }{d}_D(u,v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^k$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^k$ is distance-colored if each arc (u, v) of $D^k$ is assigned the color i where $i{=}^{\to }{d}_D(u,v)$. The digraph $D^k$ is Hamiltonian-colored...

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, the exact formulae for the Harary indices of tensor product G × Km0,m1,...,mr−1 and the strong product G⊠Km0,m1,...,mr−1 , where Km0,m1,...,mr−1 is the complete multipartite graph with partite sets of sizes m0,m1, . . . ,mr−1 are obtained. Also upper bounds for the Harary indices of tensor and strong products of graphs are estabilished. Finally, the exact formula...

It is shown that for a metric space (M,d) the following are equivalent: (i) M is a complete ℝ-tree; (ii) M is hyperconvex and has unique metric segments.