Fazal Hayat
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Research Interests
Fazal Hayat has a total of 5 co-authors. Here are the top co-authors by number of shared publications:
- Shou-Jun Xu
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, China
8 shared publications - Bo Zhou
School of Mathematical Sciences South China Normal University 510631 Guangzhou P.R. China
3 shared publications - Daniele Ettore Otera
Institute of Data Science and Digital Technologies, Vilnius University, Vilnius, Lithuania
2 shared publications - Qaisar Farhad
School of Mathematics and Statistics, Lanzhou University, Lanzhou, China
1 shared publication - Xuli Qi
Hebei Univ Sci & Technol, Dept Math, Shijiazhuang, Hebei, Peoples R China
1 shared publication
Latest Publications
- 3 March 2026
- Published by Springer Nature in Computational and Applied Mathematics
- Vol. 45 (7) , 271
On graphs with second largest closeness eigenvalue at most zero
- 3 March 2026
- Published by Springer Nature in Computational and Applied Mathematics
- Vol. 45 (7) , 271
On graphs with second largest closeness eigenvalue at most zero
- 17 June 2025
- Published by Springer Nature in Journal of the Operations Research Society of China
- p. 1-13
Eulerian Graphs with Extremal Closeness
- 17 June 2025
- Published by Springer Nature in Journal of the Operations Research Society of China
- p. 1-13
Eulerian Graphs with Extremal Closeness
- Open Access
Extremal k-Connected Graphs with Maximum Closeness
Closeness is a measure that quantifies how quickly information can spread from a given node to all other nodes in the network, reflecting the efficiency of communication within the network by indicating how close a node is to all other nodes. For a graph G, the subset S of vertices of is called vertex cut of G if the graph becomes disconnected. The minimum cardinality of S for which is either disconnected or contains precisely one vertex is called connectivity of G. A graph is called k-connected if it stays connected even when any set of fewer than k vertices is removed. In communication networks, a k-connected graph improves network reliability; even if up to nodes fail, the network remains operational, maintaining connectivity between devices. This paper aims to study the concept of closeness within n-vertex graphs with fixed connectivity. First, we identify the graphs that maximize the closeness among all graphs of order n with fixed connectivity k. Then, we determine the graphs that achieve the maximum closeness within all k-connected graphs of order n, given specific fixed parameters such as diameter, independence number, and minimum degree.Open AccessExtremal k-Connected Graphs with Maximum Closeness
Closeness is a measure that quantifies how quickly information can spread from a given node to all other nodes in the network, reflecting the efficiency of communication within the network by indicating how close a node is to all other nodes. For a graph G, the subset S of vertices of is called vertex cut of G if the graph becomes disconnected. The minimum cardinality of S for which is either disconnected or contains precisely one vertex is called connectivity of G. A graph is called k-connected if it stays connected even when any set of fewer than k vertices is removed. In communication networks, a k-connected graph improves network reliability; even if up to nodes fail, the network remains operational, maintaining connectivity between devices. This paper aims to study the concept of closeness within n-vertex graphs with fixed connectivity. First, we identify the graphs that maximize the closeness among all graphs of order n with fixed connectivity k. Then, we determine the graphs that achieve the maximum closeness within all k-connected graphs of order n, given specific fixed parameters such as diameter, independence number, and minimum degree. - Open Access
- 30 June 2024
- Published by MDPI AG in Mathematics
- Vol. 12 (13) , 2039
Maximizing Closeness in Bipartite Networks: A Graph-Theoretic Analysis
A fundamental aspect of network analysis involves pinpointing nodes that hold significant positions within the network. Graph theory has emerged as a powerful mathematical tool for this purpose, and there exist numerous graph-theoretic parameters for analyzing the stability of the system. Within this framework, various graph-theoretic parameters contribute to network analysis. One such parameter used in network analysis is the so-called closeness, which serves as a structural measure to assess the efficiency of a node’s ability to interact with other nodes in the network. Mathematically, it measures the reciprocal of the sum of the shortest distances from a node to all other nodes in the network. A bipartite network is a particular type of network in which the nodes can be divided into two disjoint sets such that no two nodes within the same set are adjacent. This paper mainly studies the problem of determining the network that maximize the closeness within bipartite networks. To be more specific, we identify those networks that maximize the closeness over bipartite networks with a fixed number of nodes and one of the fixed parameters: connectivity, dissociation number, cut edges, and diameter.
- 30 June 2024
- Published by MDPI AG in Mathematics
- Vol. 12 (13) , 2039
Open AccessMaximizing Closeness in Bipartite Networks: A Graph-Theoretic Analysis
A fundamental aspect of network analysis involves pinpointing nodes that hold significant positions within the network. Graph theory has emerged as a powerful mathematical tool for this purpose, and there exist numerous graph-theoretic parameters for analyzing the stability of the system. Within this framework, various graph-theoretic parameters contribute to network analysis. One such parameter used in network analysis is the so-called closeness, which serves as a structural measure to assess the efficiency of a node’s ability to interact with other nodes in the network. Mathematically, it measures the reciprocal of the sum of the shortest distances from a node to all other nodes in the network. A bipartite network is a particular type of network in which the nodes can be divided into two disjoint sets such that no two nodes within the same set are adjacent. This paper mainly studies the problem of determining the network that maximize the closeness within bipartite networks. To be more specific, we identify those networks that maximize the closeness over bipartite networks with a fixed number of nodes and one of the fixed parameters: connectivity, dissociation number, cut edges, and diameter.
- 29 June 2024
- Published by Springer Nature in Journal of Applied Mathematics and Computing
- Vol. 70 (5) , 5049-5061
Minimizing the second Zagreb eccentricity index in bipartite graphs with a fixed size and diameter
- 29 June 2024
- Published by Springer Nature in Journal of Applied Mathematics and Computing
- Vol. 70 (5) , 5049-5061
Minimizing the second Zagreb eccentricity index in bipartite graphs with a fixed size and diameter
- 29 March 2024
- Published by World Scientific Pub Co Pte Ltd in Discrete Mathematics, Algorithms and Applications
- Vol. 17 (02)
Extremal results on the Mostar index of trees with fixed parameters
For a graph [ ], the Mostar index of [ ] is the sum of [ ] over all edges [ ] of [ ], where [ ] denotes the number of vertices of [ ] that have a smaller distance in [ ] to [ ] than to [ ], and analogously for [ ]. 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.
- 29 March 2024
- Published by World Scientific Pub Co Pte Ltd in Discrete Mathematics, Algorithms and Applications
- Vol. 17 (02)
Extremal results on the Mostar index of trees with fixed parameters
For a graph [ ], the Mostar index of [ ] is the sum of [ ] over all edges [ ] of [ ], where [ ] denotes the number of vertices of [ ] that have a smaller distance in [ ] to [ ] than to [ ], and analogously for [ ]. 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.
- Open Access
- 1 January 2024
- Published by National Library of Serbia in Filomat
- Vol. 38 (28) , 9967-9981
Maximum values of the edge Mostar index in tricyclic graphs
For a graph G, the edge Mostar index of G is the sum of |mu (e|G)-mv (e|G)| over all edges e = uv of G, where mu (e|G) denotes the number of edges of G that have a smaller distance in G to u than to v, and analogously for mv (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.
- 1 January 2024
- Published by National Library of Serbia in Filomat
- Vol. 38 (28) , 9967-9981
Open AccessMaximum values of the edge Mostar index in tricyclic graphs
For a graph G, the edge Mostar index of G is the sum of |mu (e|G)-mv (e|G)| over all edges e = uv of G, where mu (e|G) denotes the number of edges of G that have a smaller distance in G to u than to v, and analogously for mv (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.
- Open Access
- 1 January 2024
- Published by National Library of Serbia in Filomat
- Vol. 38 (17) , 6291-6297
A lower bound on the Mostar index of tricyclic graphs
For a graph G, the Mostar index of G is the sum of |nu-nv| over all edges e = uv of G, where nu denotes the number of vertices of G that have a smaller distance in G to u than to v, and analogously for nv. In this paper, we obtain a lower bound for the Mostar index on tricyclic graphs and identify those graphs that attain the lower bound.
- 1 January 2024
- Published by National Library of Serbia in Filomat
- Vol. 38 (17) , 6291-6297
Open AccessA lower bound on the Mostar index of tricyclic graphs
For a graph G, the Mostar index of G is the sum of |nu-nv| over all edges e = uv of G, where nu denotes the number of vertices of G that have a smaller distance in G to u than to v, and analogously for nv. In this paper, we obtain a lower bound for the Mostar index on tricyclic graphs and identify those graphs that attain the lower bound.
- Open Access
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.Open AccessExtremal 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. - Open Access
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].
Open AccessSolution 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].