Search

The eccentric connectivity coindex of a graph G is defined as ECCI (G) = ∑ u v ∉ E (G) [ ecc G (u) + ecc G (v) ] = ∑ v ∈ V (G) ecc G (v) [ | V (G) | − 1 − d G (v) ] , where V (G) is the vertex set of G , E (G) is the edge set of G , ecc G (v) is the eccentricity of v ∈ V (G) and d G (v) is the degree of v in G. We present lower bounds on the eccentric connectivity coindex for bipartite graphs of given order and vertex connectivity, order and matching number, order and odd diameter. The extremal graphs are presented as well. To the best of our knowledge, our extremal graphs of given order and vertex connectivity are not extremal graphs for any other research problem in graph theory. [ABSTRACT FROM AUTHOR]

The zeroth-order general Randi'c index of a graph G is defined as Ra(G) = Σ v∊V(G) daG(v), where a ∊ R, V (G) is the vertex set of G and dG(v) is the degree of a vertex v in G. We obtain bounds on the zeroth-order general Randić index for trees of given order and distance k-domination number, where k ≥ 1. Lower bounds are given for 0 < a < 1 and upper bounds are given for a < 0 and a > 1. All the extremal graphs are presented which means that our bounds are the best possible. [ABSTRACT FROM AUTHOR]

Cruz, Monsalve and Rada [Extremal values of vertex-degree-based topological indices of chemical trees, Appl. Math. Comput. 380 (2020) 125281] posed an open problem to find the maximum value of the exponential second Zagreb index for chemical trees of given order. In this paper, we solve the open problem completely. [ABSTRACT FROM AUTHOR]

We obtain lower and upper bounds on general multiplicative Zagreb indices for trees with given independence number and order. Bounds on basic multiplicative Zagreb indices follow from our results. We also show that the bounds are best possible. [ABSTRACT FROM AUTHOR]

The harmonic index of a conected graph G is defined as H(G) = uv2E(G) 2 d(u)+d(v), where E(G) is the edge set of G, d(u) and d(v) are the degrees of vertices u and v, respectively. The spectral radius of a square matrix M is the maximum among the absolute values of the eigenvalues of M. Let q(G) be the spectral radius of the signless Laplacian matrix Q(G) = D(G)+A(G), where D(G) is the diagonal matrix having degrees of the vertices on the main diagonal and A(G) is the (0, 1) adjacency matrix of G. The harmonic index of a graph G and the spectral radius of the matrix Q(G) have been extensively studied. We investigate the relationship between the harmonic index of a graph G and the spectral radius of the matrix Q(G). We prove that for a connected graph G with n vertices, we have...and the bounds are best possible. [ABSTRACT FROM AUTHOR]

The main concern of this article is to present the complete classification of tetracyclic (chemical) graphs and establishing some extremal results with respect to the geometric‐arithmetic index, defined by GAG=∑xy∈EGdxdydx+dy, where dx denotes the degree of a vertex x in G. In addition, we characterize the extremal graphs of the first and second maximum values of geometric‐arithmetic index of n‐vertex (n ≥ 9) tetracyclic (chemical) graphs. [ABSTRACT FROM AUTHOR]

The inverse sum indeg index of a graph G is defined as ISI (G) = ∑ u v ∈ E (G) d (u) d (v) d (u) + d (v) , where d (u) is the degree of the vertex u. In a recent paper, Pattabiraman [Inverse sum indeg index of graphs, AKCE Int. J. Graphs Combinat. 15(2) (2018) 155–167] gave some lower and upper bounds on ISI index of all connected graphs in terms of Harmonic index, second Zagreb index and hyper Zagreb index. But some results were erroneous. In this note, we have corrected these results. [ABSTRACT FROM AUTHOR]

The sum of reciprocals of distance between any two vertices in a graph G is called the Harary index. We determine the n-vertex extremal graphs with the maximum Harary index for all bipartite graphs, a given matching number, a given vertex-connectivity, and with a given edge-connectivity, respectively. [ABSTRACT FROM AUTHOR]