Related papers: Proximity and Radius in Outerplanar Graphs with Bo…

The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type…

The spectral radius of a graph is the spectral radius of its adjacency matrix. A threshold graph is a simple graph whose vertices can be ordered as $v_1, v_2, \ldots, v_n$, so that for each $2 \le i \le n$, vertex $v_i$ is either adjacent…

For a fixed positive integer $k$ and a graph $G$, let $\lambda_k(G)$ denote the $k$-th largest eigenvalue of the adjacency matrix of $G$. In 2017, Tait and Tobin proved that the maximum $\lambda_1(G)$ among all outerplanar graphs on $n$…

Albertson has defined the irregularity of a simple undirected graph $G=(V,E)$ as $ \irr(G) = \sum_{uv\in E}|d_G(u)-d_G(v)|,$ where $d_G(u)$ denotes the degree of a vertex $u \in V$. Recently, this graph invariant gained interest in the…

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance from $v$ to any other vertex. The vertices whose eccentricity are equal to the diameter (the maximum eccentricity) of $G$ are called peripheral vertices. In trees the…

For two integers $r\geq 2$ and $h\geq 0$, the $h$-extra $r$-component connectivity of a graph $G$, denoted by $c\kappa_{r}^{h}$, is defined as the minimum number of vertices whose removal produces a disconnected graph with at least $r$…

An "edge guard set" of a plane graph $G$ is a subset $\Gamma$ of edges of $G$ such that each face of $G$ is incident to an endpoint of an edge in $\Gamma$. Such a set is said to guard $G$. We improve the known upper bounds on the number of…

The Wiener index of a connected graph is the summation of all distances between unordered pairs of vertices of the graph. In this paper, we give an upper bound on the Wiener index of a $k$-connected graph $G$ of order $n$ for integers…

The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an $n$-vertex maximal planar graph is at most $\lfloor\frac{1}{18}(n^3+3n^2)\rfloor$.…

We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon-Boppana-Friedman bound for graphs of even diameter, but is an…

In this paper, we study the maximum spectral radius of outerplanar $3$-uniform hypergraphs. Given a hypergraph $\mathcal{H}$, the shadow of $\mathcal{H}$ is a graph $G$ with $V(G)= V(\mathcal{H})$ and $E(G) = \{uv: uv \in h \textrm{ for…

Let $\lambda^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $\Delta$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow…

Let $m$ be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size $m$. After…

We investigate the bounds on algebraic connectivity of graphs subject to constraints on the number of edges, vertices, and topology. We show that the algebraic connectivity for any tree on $n$ vertices and with maximum degree $d$ is bounded…

A graph is $2$-planar if it has local crossing number two, that is, it can be drawn in the plane such that every edge has at most two crossings. A graph is maximal $2$-planar if no edge can be added such that the resulting graph remains…

The extremal eigenvalues including maximum eigenvalues and the minimum eigenvalues about outerplanar graphs are investigated in this paper. Some structural characterizations about the (edge) maximal bipartite outerplanar graphs are…

Given a polyhedron (planar, $3$-connected graph) $G$, we investigate its common neighbourhood graph con($G$). For cubic ($3$-regular) polyhedra, we show that the planarity of con($G$) depends on the number of odd faces of $G$, and on their…

This paper presents a sharp upper bound for the spectral radius of simple digraphs with described number of arcs. Further, the extremal graphs which attain the maximum spectral radius among all simple digraphs with fixed arcs are…

In a geometric graph, $G$, the \emph{stretch factor} between two vertices, $u$ and $w$, is the ratio between the Euclidean length of the shortest path from $u$ to $w$ in $G$ and the Euclidean distance between $u$ and $w$. The \emph{average…

For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix, and the distance energy is defined as the sum of the absolute values of the eigenvalues of its distance matrix. We establish lower and…