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.
Variable Neighborhood Search for Extremal Graphs 14: The AutoGraphiX 2 System
The AutoGraphiX (AGX) system for computer assisted or, for some of its functions, fully automated graph theory was developed at GERAD, Montreal since 1997. We report here on a new version (AGX 2) of that system. It contains many enhancements, as well as a new function for automated proof of simple propositions. Among other results, AGX 2 led to several hundred new conjectures, ranking from easy ones, proved automatically, to others requiring longer unassisted or partially assisted proofs, to open ones. Many examples are given, illustrating AGX 2’s functions and the results obtained.