The reverse Wiener index of a connected graph $G$ is defined as $$\Lambda \left(G\right)=\frac{1}{2}n(n-1)d-W\left(G\right),$$ where $n$ is the number of vertices, $d$ is the diameter, and $W\left(G\right)$ is the Wiener index (the sum of distances between all unordered pairs of vertices) of $G$. We determine the $n$-vertex non-starlike trees with the first four largest reverse Wiener indices for $n\ge 8$, and the $n$-vertex non-starlike non-caterpillar trees with the first four largest reverse Wiener indices for $n\ge 10$.

In this paper, we show that if the number of arcs in an oriented graph G (of order n) without directed cycles is sufficiently small (not greater than [2/3] n-1), then there exist arc disjoint embeddings of three copies of G into the transitive tournament TTₙ. It is the best possible bound.

In a graph G = (V,E), a non-empty set S ⊆ V is said to be an open packing set if no two vertices of S have a common neighbour in G. An open packing set which is not a proper subset of any open packing set is called a maximal open packing set. The minimum and maximum cardinalities of a maximal open packing set are respectively called the lower open packing number and the open packing number and are denoted by ρoL and ρo. In this paper, we present some bounds on these parameters.

In this paper it is proved that every $3$-connected planar graph contains a path on $3$ vertices each of which is of degree at most $15$ and a path on $4$ vertices each of which has degree at most $23$. Analogous results are stated for $3$-connected planar graphs of minimum degree $4$ and $5$. Moreover, for every pair of integers $n\ge 3$, $k\ge 4$ there is a $2$-connected planar graph such that every path on $n$ vertices in it has a vertex of degree $k$.

A graph G is said to be H-saturated if G is H-free i.e., (G has no subgraph isomorphic to H) and adding any new edge to G creates a copy of H in G. In 1986 L. Kászonyi and Zs. Tuza considered the following problem: for given m and n find the minimum size sat(n;Pₘ) of Pₘ-saturated graph of order n. They gave the number sat(n;Pₘ) for n big enough. We deal with similar problem for bipartite graphs.

We extend the notion of a potentially H-graphic sequence as follows. Let A and B be nonnegative integer sequences. The sequence pair S = (A,B) is said to be bigraphic if there is some bipartite graph G = (X ∪ Y,E) such that A and B are the degrees of the vertices in X and Y, respectively. If S is a bigraphic pair, let σ(S) denote the sum of the terms in A. Given a bigraphic pair S, and a fixed bipartite graph H, we say that S is potentially H-bigraphic if there is some realization of S containing...

The set of all non-increasing nonnegative integer sequences $\pi =$ ($d\left(v_1\right),d\left(v_2\right),\cdots ,d\left(v_n\right)$) is denoted by ${\mathrm{NS}}_n$. A sequence $\pi \in {\mathrm{NS}}_n$ is said to be graphic if it is the degree sequence of a simple graph $G$ on $n$ vertices, and such a graph $G$ is called a realization of $\pi$. The set of all graphic sequences in ${\mathrm{NS}}_n$ is denoted by ${\mathrm{GS}}_n$. A graphical sequence $\pi$ is potentially $H$-graphical if there is a realization of $\pi$ containing $H$ as a subgraph, while $\pi$ is forcibly $H$-graphical if every realization of $\pi$ contains $H$ as a subgraph. Let $K_k$ denote a complete...

The Sombor index $SO\left(G\right)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_G^2\left(u\right)+d_G^2\left(v\right)}$ of all edges $uv$ of $G$, where $d_G\left(u\right)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G=(V,E)$ is called a quasi-tree if there exists $u\in V\left(G\right)$ such that $G-u$ is a tree. Denote $𝒬(n,k)=\{G:G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G\left(u\right)=k\}$. We determined the minimum and the second minimum Sombor indices of all quasi-trees in $𝒬(n,k)$. Furthermore, we characterized the corresponding extremal graphs, respectively.

The eccentricity $e\left(v\right)$ of a vertex $v$ is defined as the distance to a farthest vertex from $v$. The radius of a graph $G$ is defined as a $r\left(G\right)={min}_{u\in V\left(G\right)}\left\{e\left(u\right)\right\}$. A graph $G$ is radius-edge-invariant if $r(G-e)=r\left(G\right)$ for every $e\in E\left(G\right)$, radius-vertex-invariant if $r(G-v)=r\left(G\right)$ for every $v\in V\left(G\right)$ and radius-adding-invariant if $r(G+e)=r\left(G\right)$ for every $e\in E\left(\overline{G}\right)$. Such classes of graphs are studied in this paper.