In a connected graph G, the distance between two vertices of G is the length of a shortest path b... more In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity index {\tau}(G) is the sum of eccentricities of all vertices of G. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.
Japan Journal of Industrial and Applied Mathematics, 2021
Let $$\mathcal {G}=(\mathcal {V}_{\mathcal {G}},\mathcal {E}_{\mathcal {G}})$$ G = ( V G , E G ) ... more Let $$\mathcal {G}=(\mathcal {V}_{\mathcal {G}},\mathcal {E}_{\mathcal {G}})$$ G = ( V G , E G ) be a connected graph with n vertices. The eccentricity $$ec_{\mathcal {G}}(w)$$ e c G ( w ) of a vertex w in $$\mathcal {G}$$ G is the maximum distance between w and any other vertex of $$\mathcal {G}$$ G . The total eccentricity index $$\tau (\mathcal {G})$$ τ ( G ) of $$\mathcal {G}$$ G is defined as $$\tau (\mathcal {G})=\sum \nolimits _{w\in \mathcal {V}_{\mathcal {G}}} ec_{\mathcal {G}}(w)$$ τ ( G ) = ∑ w ∈ V G e c G ( w ) . In this paper, we derive the trees with minimum and maximum total eccentricity index among the class of n -vertex trees with p pendent vertices. We also determine the trees with minimum and maximum total eccentricity index among the class of n -vertex trees with a given diameter.
Iota energy of signed digraphs is defined as sum of absolute values of imaginary parts of its eig... more Iota energy of signed digraphs is defined as sum of absolute values of imaginary parts of its eigenvalues. Extremal energy and iota energy of unicyclic signed digraphs is known. In this paper, we address the problem of finding signed digraphs with extremal iota energy among vertex-disjoint bicyclic signed digraphs of fixed order.
The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplici... more The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplicity of the eigenvalue zero in the spectrum of a graph is called nullity of the graph. Fan and Qian (2009) obtained the nullity set of
In a connected graph G, the distance between two vertices of G is the length of a shortest path b... more In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity index {\tau}(G) is the sum of eccentricities of all vertices of G. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.
Japan Journal of Industrial and Applied Mathematics, 2021
Let $$\mathcal {G}=(\mathcal {V}_{\mathcal {G}},\mathcal {E}_{\mathcal {G}})$$ G = ( V G , E G ) ... more Let $$\mathcal {G}=(\mathcal {V}_{\mathcal {G}},\mathcal {E}_{\mathcal {G}})$$ G = ( V G , E G ) be a connected graph with n vertices. The eccentricity $$ec_{\mathcal {G}}(w)$$ e c G ( w ) of a vertex w in $$\mathcal {G}$$ G is the maximum distance between w and any other vertex of $$\mathcal {G}$$ G . The total eccentricity index $$\tau (\mathcal {G})$$ τ ( G ) of $$\mathcal {G}$$ G is defined as $$\tau (\mathcal {G})=\sum \nolimits _{w\in \mathcal {V}_{\mathcal {G}}} ec_{\mathcal {G}}(w)$$ τ ( G ) = ∑ w ∈ V G e c G ( w ) . In this paper, we derive the trees with minimum and maximum total eccentricity index among the class of n -vertex trees with p pendent vertices. We also determine the trees with minimum and maximum total eccentricity index among the class of n -vertex trees with a given diameter.
Iota energy of signed digraphs is defined as sum of absolute values of imaginary parts of its eig... more Iota energy of signed digraphs is defined as sum of absolute values of imaginary parts of its eigenvalues. Extremal energy and iota energy of unicyclic signed digraphs is known. In this paper, we address the problem of finding signed digraphs with extremal iota energy among vertex-disjoint bicyclic signed digraphs of fixed order.
The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplici... more The eigenvalues of the adjacency matrix of a graph form the spectrum of the graph. The multiplicity of the eigenvalue zero in the spectrum of a graph is called nullity of the graph. Fan and Qian (2009) obtained the nullity set of
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Papers by Rashid Farooq