Related papers: Computing the inverse geodesic length in planar gr…

An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint on the boundary of the disk. We show that an outerstring graph with $n$ vertices has treewidth $O(\alpha\log n)$, where $\alpha$ denotes the…

We introduce the general Albertson irregularity index of a connected graph $G$ and define it as $A_{p}(G) =(\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}}$, where $p$ is a positive real number and $d(v)$ is the degree of the vertex $v$ in…

Given a graph $G$, a geodesic packing in $G$ is a set of vertex-disjoint maximal geodesics, and the geodesic packing number of $G$, ${\gpack}(G)$, is the maximum cardinality of a geodesic packing in $G$. It is proved that the decision…

In an oriented graph $\vec{G}$, the {\it inversion} of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endvertices in $X$. The {\it $(\leq p)$-inversion graph} of a labelled graph $G$, denoted by…

In the distance query model, we are given access to the vertex set of a $n$-vertex graph $G$, and an oracle that takes as input two vertices and returns the distance between these two vertices in $G$. We study how many queries are needed to…

The $r$-th iterated line graph $L^{r}(G)$ of a graph $G$ is defined by: (i) $L^{0}(G) = G$ and (ii) $L^{r}(G) = L(L^{(r- 1)}(G))$ for $r > 0$, where $L(G)$ denotes the line graph of $G$. The Hamiltonian Index $h(G)$ of $G$ is the smallest…

The walk distances in graphs are defined as the result of appropriate transformations of the $\sum_{k=0}^\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a graph and $t$ is a sufficiently small positive…

Effective resistance, which originates from the field of circuits analysis, is an important graph distance in spectral graph theory. It has found numerous applications in various areas, such as graph data mining, spectral graph…

For a graph G, the graph R(G) of a graph G is the graph obtained by adding a new vertex for each edge of G and joining each new vertex to both end vertices of the correspond- ing edge. Let I(G) be the set of newly added vertices. In this…

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of $G$, where $G$ is the underlying graph of $\widetilde{G}$. The positive inertia index, denoted by $p^{+}(G)$, and the negative inertia index, denoted by $n^{-}(G)$, of a…

We report on some findings concerning Connes' noncommutative distance $d$ on a weighted undirected graph $G$. Our main result is the lower bound $\ell/\Delta(G)\le d$ where $\ell$ is the geodesic distance and $\Delta(G)$ the degree of $G$.…

An $n$-vertex graph whose degree set consists of exactly $n-1$ elements is called antiregular graph. Such type of graphs are usually considered opposite to the regular graphs. An irregularity measure ($IM$) of a connected graph $G$ is a…

A geometric graph is a graph embedded in the plane with vertices at points and edges drawn as curves (which are usually straight line segments) between those points. The average transversal complexity of a geometric graph is the number of…

Hyperbolic tilings are natural infinite planar graphs where each vertex has degree $q$ and each face has $p$ edges for some $\frac1p+\frac1q<\frac12$. We study the structure of shortest paths in such graphs. We show that given a set of $n$…

The zeroth-order general Randi\'c index (usually denoted by $R_{\alpha}^{0}$) and variable sum exdeg index (denoted by $SEI_{a}$) of a graph $G$ are defined as $R_{\alpha}^{0}(G)= \sum_{v\in V(G)} (d_{v})^{\alpha}$ and $SEI_{a}(G)=…

We present a $(1+\epsilon)$-approximation algorithm running in $O(f(\epsilon)\cdot n \log^4 n)$ time for finding the diameter of an undirected planar graph with non-negative edge lengths.

We say that a vertex $v$ in a connected graph $G$ is decisive if the numbers of walks from $v$ of each length determine the graph $G$ rooted at $v$ up to isomorphism among all connected rooted graphs with the same number of vertices. On the…

The transmission of a vertex in a connected graph is the sum of distances from that vertex to all the other vertices. A connected graph is transmission irregular if any two distinct vertices have different transmissions. We present an…

Let $S$ be an inverse semigroup with zero and let $Z(S)^\times$ be its set of non-zero divisors with respect to the natural partial order $\le $ on $S$, that is, $a \in Z(S)^\times $ if there exists $b\in S\setminus\{0\}$ with $\omega(a, b)…

For a graph $G$, the Mostar index of $G$ is the sum of $|n_u(e)$ - $n_v(e)|$ over all edges $e=uv$ of $G$, where $n_u(e)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$, and analogously for…