Kinkar Chandra Das

Maximizing the sum of the squares of the degrees of a graph

Abstract Let G=(V,E) be a simple graph with n vertices, e edges, and vertex degrees d1,d2,…,dn. Also, let d1, dn be, respectively, the highest degree and the lowest degree of G and mi be the average of the degrees of the vertices adjacent to vertex vi∈V. It is proved that max {d j +m j : v j ∈V}⩽ 2e n−1 +n−2 with equality if and only if G is an Sn graph (K1,n−1⊆Sn⊆Kn) or a complete graph of order n−1 with one isolated vertex. Using the above result we establish the following upper bound for the sum of the squares of the degrees of a graph G: ∑ i=1 n d i 2 ⩽e 2e n−1 + n−2 n−1 d 1 +(d 1 −d n ) 1− d 1 n−1 with equality if and only if G is a star graph or a regular graph or a complete graph Kd1+1 with n−d1−1 isolated vertices. A comparison is made to another upper bound on ∑ i=1 n d i 2 , due to de Caen (Discrete Math. 185 (1998) 245). We also present several upper bounds for ∑ i=1 n d i 2 and determine the extremal graphs which achieve the bounds.

Some new bounds on the spectral radius of graphs

The eigenvalues of a graph are the eigenvalues of its adjacency matrix. This paper presents some upper and lower bounds on the greatest eigenvalue and a lower bound on the smallest eigenvalue.

On Laplacian energy of graphs

Abstract Let G be a graph with n vertices and m edges. Also let μ 1 , μ 2 , … , μ n − 1 , μ n = 0 be the eigenvalues of the Laplacian matrix of graph G . The Laplacian energy of the graph G is defined as L E = L E ( G ) = ∑ i = 1 n | μ i − 2 m n | . In this paper, we present some lower and upper bounds for L E of graph G in terms of n , the number of edges m and the maximum degree Δ . Also we give a Nordhaus–Gaddum-type result for Laplacian energy of graphs. Moreover, we obtain a relation between Laplacian energy and Laplacian-energy-like invariant of graphs.

The multiplicative Zagreb indices of graph operations

AbstractRecently, Todeschini et al. (Novel Molecular Structure Descriptors - Theory and Applications I, pp. 73-100, 2010), Todeschini and Consonni (MATCH Commun. Math. Comput. Chem. 64:359-372, 2010) have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: ∏1=∏1(G)=∏v∈V(G)dG(v)2,∏2=∏2(G)=∏uv∈E(G)dG(u)dG(v). These two graph invariants are called multiplicative Zagreb indices by Gutman (Bull. Soc. Math. Banja Luka 18:17-23, 2011). In this paper the upper bounds on the multiplicative Zagreb indices of the join, Cartesian product, corona product, composition and disjunction of graphs are derived and the indices are evaluated for some well-known graphs.MSC:05C05, 05C90, 05C07.

Relationship between the eccentric connectivity index and Zagreb indices

Abstract For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M 2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. If G is a connected graph with vertex set V ( G ) , then the eccentric connectivity index of G , ξ C ( G ) , is defined as, ∑ v i ∈ V ( G ) d i e i , where d i is the degree of a vertex v i and e i is its eccentricity. In this report we compare the eccentric connectivity index ( ξ C ) and the Zagreb indices ( M 1 and M 2 ) for chemical trees. Moreover, we compare the eccentric connectivity index ( ξ C ) and the first Zagreb index ( M 1 ) for molecular graphs.

On the Harary index of graph operations

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, expressions for the Harary indices of the join, corona product, Cartesian product, composition and disjunction of graphs are derived and the indices for some well-known graphs are evaluated. In derivations some terms appear which are similar to the Harary index and we name them the second and third Harary index.MSC:05C05, 05C07, 05C90.

Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In [5], Cvetkovic et al. (2007) have given conjectures on signless Laplacian eigenvalues of G (see also Aouchiche and Hansen (2010) [1], Oliveira et al. (2010) [14]). Here we prove two conjectures.

Estimating the Szeged index

Abstract Lower and upper bounds on Szeged index of connected (molecular) graphs are established as well as Nordhaus–Gaddum-type results, relating the Szeged index of a graph and of its complement.

Extremal t-apex trees with respect to matching energy

The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. For any integer t≥1, a graph G is called t-apex tree if there exists a t-set X⊆V(G) such that G − X is a tree, while for any Y⊆V(G) with |Y|<t, G − Y is not a tree. Let Tt(n) be the set of t-apex trees of order n. In this article, we determine the extremal graphs from Tt(n) with minimal and maximal matching energies, respectively. Moreover, as an application, the extremal cacti of order n and with s cycles have been completely characterized at which the minimal matching energy are attained.

On the Wiener polarity index of graphs

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u,?v} in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in Y. Zhang, Y. Hu, 2016 38. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus-Gaddum-type results for Wp(G). Our lower bound on W p ( G ) + W p ( G ? ) is always better than the previous lower bound given by Zhang and Hu.