MUHAMMAD KAMRAN JAMIL | Government College University, Lahore (original) (raw)

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Papers by MUHAMMAD KAMRAN JAMIL

Let G be a connected graph, and let D[G] denote the double graph of G. In this paper, we first de... more Let G be a connected graph, and let D[G] denote the double graph of G. In this paper, we first derive closed-form formulas for some distance based topological indices for D[G] in terms of G. Finally, these formulas are applied for several special kinds of graphs, such as, the complete graph, the path and the cycle. Plavsić [7] et al. and Ivanciuc et al. [4] independently introduced the Harary index in honor of Frank Harary. The Harary index is obtained

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In this paper, we show that in the class of graphs of order n and given (vertex or edge) connecti... more In this paper, we show that in the class of graphs of order n and given (vertex or edge) connectivity equal to k (or at most equal to k), 1 ≤ k ≤ n − 1, the graph K k + (K 1 ∪ K n−k−1) is the unique graph such that zeroth-order general Randiíndex , general sum-connectivity index and general Randi´onnectivity index are maximum and general hyper-Wiener index is minimum provided α ≥ 1. Also, for 2-connected (or 2-edge connected) graphs and α > 0 the unique graph minimizing these indices is the n-vertex cycle C n .

MATCH Communications in Mathematical and in Computer Chemistry

Das, Xu and Gutman [MATCH Commun. Math. Comput. Chem. 70(2013) 301-314] proved that in the class ... more Das, Xu and Gutman [MATCH Commun. Math. Comput. Chem. 70(2013) 301-314] proved that in the class of trees of order n and independence number s, the spur S n,s maximizes both first and second Zagreb indices and this graph is unique with these properties. In this paper, we show that in the same class of trees T , S n,s is the unique graph maximizing zeroth-order general Randiíndex 0 R α (T) for α > 1 and general sum-connectivity index χ α (T) for α ≥ 1. This property does not hold for general Randiíndex R α (T) if α ≥ 2 .

Let G be a connected graph, and let D[G] denote the double graph of G. In this paper, we rst der... more Let G be a connected graph, and let D[G] denote the double
graph of G. In this paper, we rst derive closed-form formulas for some
distance based topological indices for D[G] in terms of G. Finally, these formulas
are applied for several special kinds of graphs, such as, the complete
graph, the path and the cycle.

The Narumi-Katayama index NK(G) and first multiplicative Zagreb index ( ) 1 Õ G of a graph G are ... more The Narumi-Katayama index NK(G) and first multiplicative Zagreb index ( ) 1
Õ G of a
graph G are defined as the product of the degrees of the vertices of G and the product of square of
the degrees of the vertices of G , respectively. The second multiplicative Zagreb index is defined as
( )
( )
( ) ( ) 2
=
uv E G
G d u d v
Î
Õ Õ . In this paper, we compute the extremal NK(G) , ( ) 1
Õ G and
( ) 2
Õ G for the graphs with given order, number of pendant vertices and cyclomatic number.

The rst Zagreb index M1 of a graph G is equal to the sum of squares of degrees of the vertices o... more The rst Zagreb index M1 of a graph G is equal to the sum of squares of degrees of the
vertices of G. Goubko proved that for trees with n1 pendent vertices, M1 9 n1 􀀀 16. We show how
this result can be extended to hold for any connected graph with cyclomatic number
0. In addition,
graphs with n vertices, n1 pendent vertices, cyclomatic number
, and minimal M1 are characterized.
Explicit expressions for minimal M1 are given for
= 0; 1; 2, which directly can be extended for
> 2.

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MATCH Communications in Mathematical and in Computer Chemistry