Let G be a connected graph and Q(G) the signless Laplacian of G. The principal ratio γ(G) is maximum minimum entries Perron vector Q(G). In this paper, we consider among all graphs order n, show that for sufficiently large n extremal kite obtained by identifying an end vertex path to any complete graph.

For any simple connected undirected graph, it is well known that the Kirchhoff and multiplicative degree-Kirchhoff indices can be computed using the Laplacian matrix. We show that the same is true for the additive degree-Kirchhoff index and give a compact Matlab program that computes all three Kirchhoffian indices with the Laplacian matrix as the only input.

The PageRank algorithm perturbs the adjacency matrix defined by a set of web pages and hyperlinks such that the resulting matrix is positive and row-stochastic. Applying the Perron-Frobenius theorem establishes that the eigenvector associated with the dominant eigenvalue exists and is unique. For some graphs, the PageRank algorithm may yield a canonical isomorph. We propose a ranking method bas...

Given V a finite set, a self–adjoint operator on C(V ), K, is called elliptic if it is positive semi–definite and its lowest eigenvalue is simple. Therefore, there exists a unique, up to sign, unitary function ω ∈ C(V ) satisfying K(ω) = λω and then K is named (λ, ω)–elliptic. Clearly, a (λ, ω)–elliptic operator is singular iff λ = 0. Examples of elliptic operators are the so–called Schrödinger...

Abstract The Estrada index of a graph/network is defined as the trace adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many these extensions, including new ones, Gaussian, Mittag–Leffler and Onsager ones. More importantly, contextualize all indices in physico-mathematical frameworks w...