In [3], the present author used a binary operation as a tool for characterizing geodetic graphs. In this paper a new proof of the main result of the paper cited above is presented. The new proof is shorter and simpler.

Two decades ago, resistance distance was introduced to characterize “chemical distance” in (molecular) graphs. In this paper, we consider three resistance distance-based graph invariants, namely, the Kirchhoff index, the additive degree-Kirchhoff index, and the multiplicative degree-Kirchhoff index. Some Nordhaus-Gaddum-type results for these three molecular structure descriptors are obtained. In addition, a relation between these Kirchhoffian indices is established.

The known relation between the standard radius and diameter holds for graphs, but not for digraphs. We show that no upper estimation is possible for digraphs. We also give some remarks on distances, which are either metric or non-metric.

We deal with the graph operator $\overline{Pow₂}$ defined to be the complement of the square of a graph: $\overline{Pow₂}\left(G\right)=\overline{Pow₂\left(G\right)}$. Motivated by one of many open problems formulated in [6] we look for graphs that are 2-periodic with respect to this operator. We describe a class of bipartite graphs possessing the above mentioned property and prove that for any m,n ≥ 6, the complete bipartite graph $K_{m,n}$ can be decomposed in two edge-disjoint factors from . We further show that all the incidence graphs of Desarguesian finite projective geometries...

The Robinson-Foulds (RF) distance is the most popular method of evaluating the dissimilarity between phylogenetic trees. In this paper, we define and explore in detail properties of the Matching Cluster (MC) distance, which can be regarded as a refinement of the RF metric for rooted trees. Similarly to RF, MC operates on clusters of compared trees, but the distance evaluation is more complex. Using the graph theoretic approach based on a minimum-weight perfect matching in bipartite graphs, the values...

The betweenness centrality of a vertex of a graph is the fraction of shortest paths between all pairs of vertices passing through that vertex. In this paper, we study properties and constructions of graphs whose vertices have the same value of betweenness centrality (betweenness-uniform graphs); we show that this property holds for distance-regular graphs (which include strongly regular graphs) and various graphs obtained by graph cloning and local join operation. In addition, we show that, for...

For an ordered $k$-decomposition $𝒟=\{G_1,G_2,\cdots ,G_k\}$ of a connected graph $G$ and an edge $e$ of $G$, the $𝒟$-code of $e$ is the $k$-tuple $c_{𝒟}\left(e\right)=(d(e,G_1),d(e,G_2),...,d(e,G_k))$, where $d(e,G_i)$ is the distance from $e$ to $G_i$. A decomposition $𝒟$ is resolving if every two distinct edges of $G$ have distinct $𝒟$-codes. The minimum $k$ for which $G$ has a resolving $k$-decomposition is its decomposition dimension ${dim}_d\left(G\right)$. A resolving decomposition $𝒟$ of $G$ is connected if each $G_i$ is connected for $1\le i\le k$. The minimum $k$ for which $G$ has a connected resolving $k$-decomposition is its connected decomposition...

The concepts of critical and cocritical radius edge-invariant graphs are introduced. We prove that every graph can be embedded as an induced subgraph of a critical or cocritical radius-edge-invariant graph. We show that every cocritical radius-edge-invariant graph of radius r ≥ 15 must have at least 3r+2 vertices.

The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number ${\chi }_D\left(G\right)$ of G. Extending these concepts to infinite graphs we prove that $D\left(Q_{\aleph }₀\right)=2$ and ${\chi }_D\left(Q_{\aleph }₀\right)=3$, where $Q_{\aleph }₀$ denotes the hypercube of countable dimension. We also show that ${\chi }_D\left(Q₄\right)=4$, thereby completing the investigation of finite hypercubes with respect to ${\chi }_D$. Our results...

For two vertices u and v in a graph G = (V,E), the detour distance D(u,v) is the length of a longest u-v path in G. A u-v path of length D(u,v) is called a u-v detour. A set S ⊆V is called an edge detour set if every edge in G lies on a detour joining a pair of vertices of S. The edge detour number dn₁(G) of G is the minimum order of its edge detour sets and any edge detour set of order dn₁(G) is an edge detour basis of G. A connected graph G is called an edge detour graph if it has an edge detour...

We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link $L\left(v_i\right)=G_i$ for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.

In his PhD thesis [Structural aspects of switching classes, Leiden Institute of Advanced Computer Science, 2001] Hage posed the following problem: “characterize the maximum size graphs in switching classes”. These are called s-maximal graphs. In this paper, we study the properties of such graphs. In particular, we show that any graph with sufficiently large minimum degree is s-maximal, we prove that join of two s-maximal graphs is also an s-maximal graph, we give complete characterization of triangle-free...