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 + √

The Wiener index of a connected graph is the sum distances between all unordered pairs vertices. A Eulerian if its vertex degrees are even. In Gutman et al. (2014) authors proved that cycle unique maximising among graphs given order. They also conjectured for order n≥26 consisting on n−2 vertices and triangle share with second largest index. conjecture known to hold n≤25 exception six values. t...

‎let $g=(v(g),e(g))$ be a simple connected graph with vertex set $v(g)$ and edge‎ ‎set $e(g)$‎. ‎the (first) edge-hyper wiener index of the graph $g$ is defined as‎: ‎$$ww_{e}(g)=sum_{{f,g}subseteq e(g)}(d_{e}(f,g|g)+d_{e}^{2}(f,g|g))=frac{1}{2}sum_{fin e(g)}(d_{e}(f|g)+d^{2}_{e}(f|g)),$$‎ ‎where $d_{e}(f,g|g)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $e(g)$ and $d_{e}(f|g)=s...

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.

The terminal Wiener index TW = TW (G) of a graph G is equal to the sum of distances between all pairs of pendent vertices of G . This distance–based molecular structure descriptor was put forward quite recently [I. Gutman, B. Furtula, M. Petrović, J. Math. Chem. 46 (2009) 522–531]. In this paper we report results on TW of thorn graphs. Also a method for calculation of TW of dendrimers is descri...