Related papers: On the difference between the Szeged and Wiener in…

The edge Szeged index and edge-vertex Szeged index of a graph are defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$ and $Sz_{ev}(G)=\frac{1}{2} \sum\limits_{uv \in E(G)}[n_{u}(uv|G)m_{v}(uv|G)+n_{v}(uv|G)m_{u}(uv|G)],$…

The celebrated K\H{o}v\'ari-S\'os-Tur\'an theorem states that any $n$-vertex graph containing no copy of the complete bipartite graph $K_{s,s}$ has at most $O_s(n^{2-1/s})$ edges. In the past two decades, motivated by the applications in…

The first Zagreb index of a graph $G$ is the sum of squares of the vertex degrees in a graph and the second Zagreb index of $G$ is the sum of products of degrees of adjacent vertices in $G$. The imbalance of an edge in $G$ is the numerical…

Affirming a conjecture of Erd\H{o}s and Renyi we prove that for any (real number) c_1>0 for some c_2>0, if a graph G has no c_1(log n) nodes on which the graph is complete or edgeless (i.e. G exemplifies |G| not-> (c_1 log n)^2_2) then G…

The first multiplicative Zagreb index of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the…

A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least 5 or the complement of one.…

This paper considers the \textit{Zarankiewicz problem} in graphs with low-dimensional geometric representation (i.e., low Ferrers dimension). Our first result reveals a separation between bipartite graphs of Ferrers dimension three and…

Karo\'nski, {\L}uczak and Thomason conjectured in 2004 that for every finite graph without isolated edge, the edges can be assigned weights from $\{1,2,3\}$ in such a way that the endvertices of each edge have different sums of incident…

Topological indices are parameters associated with graphs that have many applications in different areas such as mathematical chemistry. Among various topological indices, the Wiener index is classical \cite{w}. In this paper, we prove a…

The Wiener index W(G) of a connected graph $G$ is the sum of distances between all pairs of vertices in G$. In this paper, we first give the recurrences or explicit formulae for computing the Wiener indices of spiro and polyphenyl hexagonal…

Bondy and Vince proved that a graph of minimum degree at least three contains two cycles whose lengths differ by one or two, which was conjectured by Erd\H{o}s. Gao, Li, Ma and Xie gave an average degree counterpart of Bondy-Vince's result,…

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $$ {\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}} $$ (where…

We prove that the Wiener Index $W(G)$ of a Maximal Planar graph $G$ with $n$ vertices satisfies $W(G) \leq \Big{\lfloor} \frac{1}{18}(n^3 + 3n^2) \Big{\rfloor}$ for $3 \leq n \leq 18$.

We give an upper bound on the number of perfect matchings in simple graphs with a given number of vertices and edges. We apply this result to give an upper bound on the number of 2-factors in a directed complete bipartite balanced graph on…

In the paper we develop new methods for calculating the two well-known topological indices, the degree-distance and the Gutman index. Firstly, we prove that the Wiener index of a double vertex-weighted graph can be computed from the Wiener…

Twin-width is a recently introduced graph parameter for finite graphs. It is an open problem to determine whether there is an $n$-vertex graph having twin-width at least $n/2$ (due to J. Ahn, K. Hendrey, D. Kim and S. Oum). In an earlier…

We complement our study of 2-connectivity in directed graphs, by considering the computation of the following 2-vertex-connectivity relations: We say that two vertices v and w are 2-vertex-connected if there are two internally…

Let $S$ be a set of vertices of a connected graph $G$. The Steiner distance of $S$ is the minimum size of a connected subgraph of $G$ containing all the vertices of $S$. The sum of all Steiner distances on sets of size $k$ is called the…

Motivated by recent extensive studies on Wenger graphs, we introduce a new infinite class of bipartite graphs of the similar type, called linearized Wenger graphs. The spectrum, diameter and girth of these linearized Wenger graphs are…

Fullerene graphs are mathematical models of fullerene molecules. The Wiener $(r,s)$-complexity of a fullerene graph $G$ with vertex set $V(G)$ is the number of pairwise distinct values of $(r,s)$-transmission $tr_{r,s}(v)$ of its vertices…