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Papers by Patrick Ali
Graph Theory [Working Title], 2021
Graph invariants such as distance have a wide application in life, in particular when networks re... more Graph invariants such as distance have a wide application in life, in particular when networks represent scenarios in form of either a bipartite or non-bipartite graph. Average distance μ of a graph G is one of the well-studied graph invariants. The graph invariants are often used in studying efficiency and stability of networks. However, the concept of average distance in a neighborhood graph G′ and its application has been less studied. In this chapter, we have studied properties of neighborhood graph and its invariants and deduced propositions and proofs to compare radius and average distance measures between G and G′. Our results show that if G is a connected bipartite graph and G′ its neighborhood, then radG1′≤radG and radG2′≤radG whenever G1′ and G2′ are components of G′. In addition, we showed that radG′≤radG for all r≥1 whenever G is a connected non-bipartite graph and G′ its neighborhood. Further, we also proved that if G is a connected graph and G′ its neighborhood, then a...
We prove that if G is a 3-connected plane graph of order p, maximum face length l and radius rad(... more We prove that if G is a 3-connected plane graph of order p, maximum face length l and radius rad(G), then the bound rad(G) ≤ p 6 + 5l 6 + 2 3 holds. For constant l, our bound is shown to be asymptotically sharp and improves on a bound by Harant (1990) [6]. Furthermore we extend these results to 4-and 5-connected planar graphs.
Graph Theory [Working Title], 2021
Graph invariants such as distance have a wide application in life, in particular when networks re... more Graph invariants such as distance have a wide application in life, in particular when networks represent scenarios in form of either a bipartite or non-bipartite graph. Average distance μ of a graph G is one of the well-studied graph invariants. The graph invariants are often used in studying efficiency and stability of networks. However, the concept of average distance in a neighborhood graph G′ and its application has been less studied. In this chapter, we have studied properties of neighborhood graph and its invariants and deduced propositions and proofs to compare radius and average distance measures between G and G′. Our results show that if G is a connected bipartite graph and G′ its neighborhood, then radG1′≤radG and radG2′≤radG whenever G1′ and G2′ are components of G′. In addition, we showed that radG′≤radG for all r≥1 whenever G is a connected non-bipartite graph and G′ its neighborhood. Further, we also proved that if G is a connected graph and G′ its neighborhood, then a...
We prove that if G is a 3-connected plane graph of order p, maximum face length l and radius rad(... more We prove that if G is a 3-connected plane graph of order p, maximum face length l and radius rad(G), then the bound rad(G) ≤ p 6 + 5l 6 + 2 3 holds. For constant l, our bound is shown to be asymptotically sharp and improves on a bound by Harant (1990) [6]. Furthermore we extend these results to 4-and 5-connected planar graphs.