arXiv subjects

Fazal Hayat

Publications and source records attributed to Fazal Hayat.

6 recordsLinked to original sources

arXiv2024

Maximum values of the edge Mostar index in tricyclic graphs

For a graph $G$, the edge Mostar index of $G$ is the sum of $|m_u(e|G)-m_v(e|G)|$ over all edges $e=uv$ of $G$, where $m_u(e|G)$ denotes the number of edges of $G$ that have a smaller distance in $G$ to $u$ than to $v$, and analogously for $m_v(e|G)$. This paper mainly studies the problem of determining the graphs that maximize the edge Mostar index among tricyclic graphs. To be specific, we determine a sharp upper bound for the edge Mostar index on tricyclic graphs and identify the graphs that attain the bound.

math.CO↗

arXiv2024

Disproof of a conjecture on the edge Mostar index

For a given connected graph $G$, the edge Mostar index $Mo_e(G)$ is defined as $Mo_e(G)=\sum_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|$, where $m_u(e|G)$ and $m_v(e|G)$ are respectively, the number of edges of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of edges of $G$ lying closer to vertex $v$ than to vertex $u$. We determine a sharp upper bound for the edge Mostar index on bicyclic graphs and identify the graphs that attain the bound, which disproves a conjecture proposed by Liu et al. [Iranian J. Math. Chem. 11(2) (2020) 95--106].

math.CO↗

arXiv2023

Solution to an open problem on the closeness of graphs

A network can be analyzed by means of many graph theoretical parameters. In the context of networks analysis, closeness is a structural metric that evaluates a node's significance inside a network. A cactus is a connected graph in which any block is either a cut edge or a cycle. This paper analyzes the closeness of cacti, we determine the unique graph that minimizes the closeness over all cacti with fixed numbers of vertices and cycles, which solves an open problem proposed by Poklukar \& Žerovnik [Fundam. Inform. 167 (2019) 219--234].

cs.SI↗

arXiv2023

Extremal results on the Mostar index of trees with fixed parameters

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 $n_v(e)$. We determine all the graphs that maximize and minimize the Mostar index respectively over all trees in terms of some fixed parameters like the number of odd vertices, the number of vertices of degree two, and the number of pendent paths of fixed length.

math.CO↗

arXiv2019

On the maximum CEI of graphs with paprameters

The connective eccentricity index (CEI) of a graph $G$ is defined as $ξ^{ce}(G)=\sum_{v \in V(G)}\frac{d_G(v)}{\varepsilon_G(v)}$, where $d_G(v)$ is the degree of $v$ and $\varepsilon_G(v)$ is the eccentricity of $v$. In this paper, we characterize the unique graphs with maximum CEI from three classes of graphs: the $n$-vertex graphs with fixed connectivity and diameter, the $n$-vertex graphs with fixed connectivity and independence number, and the $n$-vertex graphs with fixed connectivity and minimum degree.

math.CO↗

arXiv2019

On extremal results of multiplicative Zagreb indices of trees with given distance $k$-domination number

The first multiplicative Zagreb index $Π_1$ of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index $Π_2$ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for $Π_1$ and upper bound for $Π_2$ of trees with given distance $k$-domination number, and characterize those trees attaining the bounds.

math.CO↗