Related papers: Note on the spectra of Steiner distance hypermatri…

The distance of a vertex in a graph is the sum of distances from that vertex to all other vertices of the graph. The Wiener index of a graph is the sum of distances between all its unordered pairs of vertices. A graph has been obtained that…

A vertex of degree one in a tree is called an end vertex and a vertex of degree at least three is called a branch vertex. For a graph $G$, let $\sigma_2$ be the minimum degree sum of two nonadjacent vertices in $G$. We consider tree…

Mader proved that for $k\geq 2$ and $n\geq 2k$, every $n$-vertex graph with no $(k+1)$-connected subgraphs has at most $(1+\frac{1}{\sqrt{2}})k(n-k)$ edges. He also conjectured that for $n$ large with respect to $k$, every such graph has at…

For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…

The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least 2 and $S\subseteq V(G)$, the Steiner distance…

A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this order.…

Let $A(G)$ be the adjacency matrix of a graph $G$ with $\lambda_{1}(G)$, $\lambda_{2}(G)$, ..., $\lambda_{n}(G)$ being its eigenvalues in non-increasing order. Call the number $S_k(G):=\sum_{i=1}^{n}\lambda_{i}^k(G) (k=0,1,...,n-1)$ the…

The geometric bottleneck Steiner network problem on a set of vertices $X$ embedded in a normed plane requires one to construct a graph $G$ spanning $X$ and a variable set of $k\geq 0$ additional points, such that the length of the longest…

We present sublinear-time (randomized) algorithms for finding simple cycles of length at least $k\geq 3$ and tree-minors in bounded-degree graphs. The complexity of these algorithms is related to the distance of the graph from being…

Let $G$ be a graph. The Steiner distance of $W\subseteq V(G)$ is the minimum size of a connected subgraph of $G$ containing $W$. Such a subgraph is necessarily a tree called a Steiner $W$-tree. The set $A\subseteq V(G)$ is a $k$-Steiner…

The Steiner $k$-eccentricity of a vertex $v$ of a graph $G$ is the maximum Steiner distance over all $k$-subsets of $V (G)$ which contain $v$. In this note, we design a linear algorithm for computing the Steiner $3$-eccentricities and the…

The Wiener index (the distance) of a connected graph is the sum of distances between all pairs of vertices. In this paper, we study the maximum possible value of this invariant among graphs on $n$ vertices with fixed number of blocks $p$.…

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge…

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem…

The metric dimension, $\dim(G)$, of a graph $G$ is a graph parameter motivated by robot navigation that has been studied extensively. Let $G$ be a graph with vertex set $V(G)$, and let $d(x,y)$ denote the length of a shortest $x-y$ path in…

Let $k\geq2$ be an integer. A tree $T$ is called a $k$-tree if $d_T(v)\leq k$ for each $v\in V(T)$, that is, the maximum degree of a $k$-tree is at most $k$. Let $\lambda_1(D(G))$ denote the distance spectral radius in $G$, where $D(G)$…

Let $c(G)$ denote the circumference of a graph $G$, i.e., the number of vertices in its longest cycle. For positive integers $n$ and $k$ with $n>k$, let $\varGamma(n;k)$ be the class of graphs of order $n$ with $c(G) = n-k$ such that every…

Let $\mathcal{H}$ be a $k$-uniform hypergraph. A chain in $\mathcal{H}$ is a sequence of its vertices such that every $k$ consecutive vertices form an edge. In 1999 Katona and Kierstead suggested to use chains in hypergraphs as the…

Let $D(G)$ be the distance matrix of a simple connected graph $G$. The Hadamard product $D(G)~\circ~ D(G)$ is called the squared distance matrix of $G$, and is denoted by $\Delta(G)$. A simple connected graph is called a starlike block…

A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on…