The Wiener indexW (G) of a connected graphG is defined to be the sum ∑ u,v d(u, v) of the distances between the pairs of vertices in G. Similarly, the edge-Wiener index We(G) of G is defined to be the sum ∑ e,f d(e, f) of the distances between the pairs of edges in G, or equivalently, the Wiener index of the line graph L(G). Finally, the Gutman index Gut(G) is defined to be the sum ∑ u,v deg(u)...

In chemical graph theory, distance-degree-based topological indices are expressions of the form ∑ u6=v F (deg(u), deg(v)), d(u, v)), where F is a function, deg(u) the degree of u, and d(u, v) the distance between u and v. Setting F to be (deg(u) + deg(v))d(u, v), deg(u)deg(v)d(u, v), (deg(u)+deg(v))d(u, v)−1, and deg(u)deg(v)d(u, v)−1, we get the degree distance index DD, the Gutman index Gut, ...

In a recent publication in the journal Bioinformatics, Boycheva et al. 6 describe an empirical predictor for the translational step-times of codon pairs that differs from an experimentally verified statistically-based predictor previously described by Gutman and Hatfield. Here we review the statistical and experimental foundations for the Gutman and Hatfield conclusions and compare them to thos...

Topological indices are a class of numerical invariants that predict certain physical and chemical properties molecules. Recently, two novel topological indices, named as Sombor index reduced index, were introduced by Gutman, defined where denotes the degree vertex in . In this paper, our aim is to order trees, unicyclic graphs, bicyclic graphs tricyclic with respect index. We determine first f...

Given a complex m × n matrix A, we index its singular values as σ1 (A) ≥ σ2 (A) ≥ ... and call the value E (A) = σ1 (A)+σ2 (A)+ ... the energy of A, thereby extending the concept of graph energy, introduced by Gutman. Let 2 ≤ m ≤ n, A be an m×n nonnegative matrix with maximum entry α, and ‖A‖ 1 ≥ nα. Extending previous results of Koolen and Moulten for graphs, we prove that E (A) ≤ ‖A‖1 √ mn + √

Let G be a graph of order n and let Λ(G, λ) = ∑n k=0(−1)ckλ be the characteristic polynomial of its Laplacian matrix. Zhou and Gutman recently proved that among all trees of order n, the kth coefficient ck is largest when the tree is a path, and is smallest for stars. A new proof and a strengthening of this result is provided. A relation to the Wiener index is discussed.

In the field of mathematics, there are many branches study, especially in graph theory, mathematically a is pair sets, which consists non-empty set whose members called vertices and distinct unordered pairs edges. One example from group coprime graph, where defined as two with different x y neighbors if only (|x|,|y|)=1. this author discusses Harmonic Index Gutman Coprime Graph Integer Group Mo...