Related papers: On the spread of outerplanar graphs

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at…

Let $G$ be an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that \Delta-\lambda_1>\frac{1}{nD} where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of…

Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup…

To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every n-vertex geometric planar graph can be untangled while…

An outerstring graph is an intersection graph of curves that lie in a common half-plane and have one endpoint on the boundary of that half-plane. We prove that the class of outerstring graphs is $\chi$-bounded, which means that their…

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing edges. It has been conjectured for a long time that for every fixed k, the maximum number of edges of a k-quasi-planar graph with n vertices is…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has a vertex (edge) labeling with $d$ labels that is preserved only by a trivial automorphism. In this paper we consider the maximal…

We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.

For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its…

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…

Borradaile, Le and Sherman-Bennett [Graphs and Combinatorics, 2017] proved that every $n$-vertex $2$-outerplane graph has a set of at least $2n/3$ vertices that induces an outerplane graph. We identify a major flaw in their proof and…

For a set of graphs $\mathcal{F}$, a graph is said to be $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. Let Ex$_{sp}(n,\mathcal{F})$ denote the graphs with the maximum spectral radius among all…

For a simple graph $F$, let $\mathrm{EX}(n, F)$ and $\mathrm{EX_{sp}}(n,F)$ be the set of graphs with the maximum number of edges and the set of graphs with the maximum spectral radius in an $n$-vertex graph without any copy of the graph…

A 1-planar graph is a graph which has a drawing on the plane such that each edge is crossed at most once. If a 1-planar graph is drawn in that way, the drawing is called a {\it 1-plane graph}. A graph is maximal 1-plane (or 1-planar) if no…

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We…

Let $\mathcal{F}$ be a set of graphs. The planar Tur\'an number, $ex_{\mathcal{P}}(n,\mathcal{F})$, is the maximum number of edges in an $n$-vertex planar graph which does not contain any member of $\mathcal{F}$ as a subgraph. In this…

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $\Delta(G)$ the maximum degree of a graph $G$ and let $\rho(G)$ be the spectral radius of $G$. In this article we present a lower…

For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$,…

Albertson and Berman conjectured that every planar graph has an induced forest on half of its vertices. The best known lower bound, due to Borodin, is that every planar graph has an induced forest on two fifths of its vertices. In a related…

In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological…