Remoteness and distance, distance (signless) Laplacian eigenvalues of a graph
Let G be a connected graph of order n. The remoteness of G, denoted by ρ, is the maximum average distance from a vertex to all other vertices. Let ∂1≥⋯≥∂n$\partial_{1}\geq\cdots\geq\partial_{n}$, ∂1L≥⋯≥∂nL$\partial_{1}^{L}\geq\cdots\geq\partial_{n}^{L}$ and ∂1Q≥⋯≥∂nQ$\partial_{1} ^{Q}\geq\cdots\geq\partial_{n}^{Q}$ be the distance, distance Laplacian and distance signless Laplacian eigenvalues of G, respectively. In this paper, we give lower bounds on ρ+∂1$\rho+\partial _{1}$, ρ−∂n$\rho-\partial_{n}$, ρ+∂1L$\rho+\partial_{1}^{L}$, ∂1L−ρ$\partial_{1} ^{L}-\rho$, 2ρ+∂1Q$2\rho+\partial_{1}^{Q}$ and ∂1Q−2ρ$\partial_{1}^{Q}-2\rho$ and the corresponding extremal graphs are also characterized.
On the least distance eigenvalue and its applications on the distance spread
Let G be a connected graph with order n and D ( G ) be its distance matrix. Suppose that λ 1 ( D ) ? ? ? λ n ( D ) are the distance eigenvalues of G . In this paper, we give an upper bound on the least distance eigenvalue and characterize all the connected graphs with - 1 - 2 ? λ n ( D ) ? a where a is the smallest root of x 3 - x 2 - 11 x - 7 = 0 and a ? ( - 1 - 2 , - 2 ) . Furthermore, we show that connected graphs with λ n ( D ) ? - 1 - 2 are determined by their distance spectra. As applications, we give some lower bounds on the distance spread of graphs with given some parameters. In the end, we characterize connected graphs with the ( k + 1 ) th smallest distance spread.
Birth-death processes with killing
The purpose of this note is to point out that Karlin and McGregor's integral representation for the transition probabilities of a birth-death process on a semi-infinite lattice with an absorbing bottom state remains valid if one allows the possibility of absorption into the bottom state from any other state. Conditions for uniqueness of the minimal transition function are also given.