We show positivity of the Q-matrix of four kinds of graph products: direct product (Cartesian product), star product, comb product, and free product. During the discussion we give an alternative simple proof of the Markov product theorem on positive definite kernels.

A buttoning of a tree that has vertices v1, v2, . . . , vn is a closed walk that starts at v1 and travels along the shortest path in the tree to v2, and then along the shortest path to v3, and so forth, finishing with the shortest path from vn to v1. Inspired by a problem about buttoning a shirt inefficiently, we determine the maximum length of buttonings of trees

For a connected graph $G$ of order $n\ge 3$ and an ordering $s{v}_1$, $v_2,\cdots ,v_n$ of the vertices of $G$, $d\left(s\right)={\sum }_{i=1}^{n-1}d(v_i,v_{i+1})$, where $d(v_i,v_{i+1})$ is the distance between $v_i$ and $v_{i+1}$. The traceable number $t\left(G\right)$ of $G$ is defined by $t\left(G\right)=min\left\}d\left(s\right)\right\{,$ where the minimum is taken over all sequences $s$ of the elements of $V\left(G\right)$. It is shown that if $G$ is a nontrivial connected graph of order $n$ such that $l$ is the length of a longest path in $G$ and $p$ is the maximum size of a spanning linear forest in $G$, then $2n-2-p\le t\left(G\right)\le 2n-2-l$ and both these bounds are sharp. We establish a formula for the traceable number of...

Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G×P₃ is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G×K₂ are proved, and it is shown that the only nonbipartite quasi-median direct product is K₃×K₃.

For any vertex v and any edge e in a non-trivial connected graph G, the distance sum d(v) of v is $d\left(v\right)={\sum }_{u\in V}d(v,u)$, the vertex-to-edge distance sum d₁(v) of v is $d₁\left(v\right)={\sum }_{e\in E}d(v,e)$, the edge-to-vertex distance sum d₂(e) of e is $d₂\left(e\right)={\sum }_{v\in V}d(e,v)$ and the edge-to-edge distance sum d₃(e) of e is $d₃\left(e\right)={\sum }_{f\in E}d(e,f)$. The set M(G) of all vertices v for which d(v) is minimum is the median of G; the set M₁(G) of all vertices v for which d₁(v) is minimum is the vertex-to-edge median of G; the set M₂(G) of all edges e for which d₂(e) is minimum is the edge-to-vertex median...

Two numerical invariants $\Delta \left(G\right)$ and $\Gamma \left(G\right)$ of a graph, related to the concept of median, are studied.

We show that superreflexivity can be characterized in terms of bilipschitz embeddability of word hyperbolic groups.We compare characterizations of superrefiexivity in terms of diamond graphs and binary trees.We show that there exist sequences of series-parallel graphs of increasing topological complexitywhich admit uniformly bilipschitz embeddings into a Hilbert space, and thus do not characterize superrefiexivity.

Zero forcing number has recently become an interesting graph parameter studied in its own right since its introduction by the “AIM Minimum Rank–Special Graphs Work Group”, whereas metric dimension is a well-known graph parameter. We investigate the metric dimension and the zero forcing number of some line graphs by first determining the metric dimension and the zero forcing number of the line graphs of wheel graphs and the bouquet of circles. We prove that $Z\left(G\right)\le 2Z\left(L\right(G\left)\right)$ for a simple and connected graph $G$. Further,...

In this paper we consider the point character of metric spaces. This parameter which is a uniform version of dimension, was introduced in the context of uniform spaces in the late seventies by Jan Pelant, Cardinal reflections and point-character of uniformities, Seminar Uniform Spaces (Prague, 1973–1974), Math. Inst. Czech. Acad. Sci., Prague, 1975, pp. 149–158. Here we prove for each cardinal $\kappa$, the existence of a metric space of cardinality and point character $\kappa$. Since the point character can...

There is a hypothesis that a non-selfcentric radially-maximal graph of radius r has at least 3r-1 vertices. Using some recent results we prove this hypothesis for r = 4.

Let V be a finite set and 𝓜 a collection of subsets of V. Then 𝓜 is an alignment of V if and only if 𝓜 is closed under taking intersections and contains both V and the empty set. If 𝓜 is an alignment of V, then the elements of 𝓜 are called convex sets and the pair (V,𝓜 ) is called an alignment or a convexity. If S ⊆ V, then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ ℳ. Then x ∈ X is an extreme point for X if X∖{x} ∈ ℳ. A convex geometry on a finite set is...

The concept of a signpost system on a set is introduced. It is a ternary relation on the set satisfying three fairly natural axioms. Its underlying graph is introduced. When the underlying graph is disconnected some unexpected things may happen. The main focus are signpost systems satisfying some extra axioms. Their underlying graphs have lots of structure: the components are modular graphs or median graphs. Yet another axiom guarantees that the underlying graph is also connected. The main results...

It is shown in this note that some matching-related properties of graphs, such as their factor-criticality, regularizability and the existence of perfect 2-matchings, are preserved when iterating Mycielski's construction.

n-ary transit functions are introduced as a generalization of binary (2-ary) transit functions. We show that they can be associated with convexities in natural way and discuss the Steiner convexity as a natural n-ary generalization of geodesicaly convexity. Furthermore, we generalize the betweenness axioms to n-ary transit functions and discuss the connectivity conditions for underlying hypergraph. Also n-ary all paths transit function is considered.

Chartrand et al. (2004) have given an upper bound for the nearly antipodal chromatic number $a{c}^{\text{'}}\left(P_n\right)$ as $\left(\genfrac{}{}{0pt}{}{n-2}{2}\right)+2$ for $n\ge 9$ and have found the exact value of $a{c}^{\text{'}}\left(P_n\right)$ for $n=5,6,7,8$. Here we determine the exact values of $a{c}^{\text{'}}\left(P_n\right)$ for $n\ge 8$. They are $2p^2-6p+8$ for $n=2p$ and $2p^2-4p+6$ for $n=2p+1$. The exact value of the radio antipodal number $ac\left(P_n\right)$ for the path $P_n$ of order $n$ has been determined by Khennoufa and Togni in 2005 as $2p^2-2p+3$ for $n=2p+1$ and $2p^2-4p+5$ for $n=2p$. Although the value of $ac\left(P_n\right)$ determined there is correct, we found a mistake in the proof of the lower bound when $n=2p$ (Theorem $6$). However,...