The Ermakov-Pinney Equation, x?? + ?2x = h2/x3, has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the ?2 and h2. The former has no effect upon the algebra of the Lie point symmetries of the equation. The latter destroys the sl(2,R) symmetry and a single symmetry persists only when there is a specific relationship between the two time-dependent functions introduced. We calculate the form of the corresponding autonomous equation for these cases.

A Roman dominating function on a graph G is a function f:V(G) → {0,1,2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function f is the sum, ΣuV(G) f(u), of the weights of the vertices. The Roman domination number is the minimum weight of a Roman dominating function in G. A total Roman domination function is a Roman dominating function with the additional property that the subgraph of G induced by the set of all vertices of positive weight has no isolated vertex. The total Roman domination number is the minimum weight of a total Roman domination function on G. We establish lower and upper bounds on the total Roman domination number. We relate the total Roman domination to domination parameters, including the domination number, the total domination number and Roman domination number.

In this work we show the existence and uniqueness of solutions to the\n fractional nonlinear Cauchy problem containing Katugampola derivative.\n Moreover, in the linear case, we give a formula for the solution.

Let D be a non-trivial simple t-design. In this paper, for 1 ≤ s ≤ t, we\n generalize the concept of the incidence graph of D and construct a new bi-\n partite regular graph Γ. We obtain that the edge-transitivity of the graph\n Γ is equivalent to the s-ag-transitivity of the design D. We then, for s =\n 2, classify the semisymmetric graphs among the graphs Γ constructed from\n bi-planes and triplanes. Finally, we study the connectedness and the energy\n of incidence graphs. Several open problems are proposed, one of which asks\n whether the incidence graphs have large vertex-connectivity.

We investigate a prey-predator model, which we describe by an ordinary- parabolic system. We obtain four types of wave solutions of this system, which are connecting different equilibria. To establish the existence of four types of traveling wave solutions with double wave speeds, we introduce a new approach to constructing monotonous iteration schemes. Moreover, by using spreading speeds, we establish the non-existence of traveling wave solutions. Our results provide insight into the dynamics of this model system.

A k??quasiperfect dominating set of a connected graph G is a vertex subset S such that every vertex not in S is adjacent to at least one and at most k vertices in S. The cardinality of a minimum k-quasiperfect dominating set in G is denoted by ?1k(G). These graph parameters were first introduced by Chellali et al. (2013) as a generalization of both the perfect domination number ?11(G) and the domination number ?(G). The study of the so-called quasiperfect domination chain ?11(G) ? ?12(G)?... ? ?1?(G) = ?(G) enable us to analyze how far minimum dominating sets are from being perfect. In this paper, we provide, for any tree T and any positive integer k, a tight upper bound of ?1k(T). We also prove that there are trees satisfying all possible equalities and inequalities in this chain. Finally a linear algorithm for computing ?1k(T) in any tree T is presented.

A graceful labeling of a graph with q edges is a labeling of its vertices using the integers in [0, q], such that no two vertices are assigned the same label and each edge is uniquely identified by the absolute difference between the labels of its endpoints. The well known Graceful Tree Conjecture (GTC) states that all trees are graceful, and it remains open. It was proved in 1999 by Broersma and Hoede that there is an equivalent conjecture for GTC stating that all trees containing a perfect matching are strongly graceful (graceful with an extra condition). In this paper we extend the above result by showing that there exist infinitely many equivalent versions of the GTC. Moreover we verify these infinitely many equivalent conjectures of GTC for trees of diameter at most 7. Among others we are also able to identify new graceful trees and in particular generalize the ?-construction of Stanton-Zarnke (and later Koh- Rogers-Tan) for building graceful trees through two smaller given graceful trees.

Let G be an n-vertex (n ? 3) simple graph embeddable on a surface of Euler genus (the number of crosscaps plus twice the number of handles). In this paper, we present upper bounds for the signless Laplacian spectral radius of planar graphs, outerplanar graphs and Halin graphs, respectively, in terms of order and maximum degree. We also demonstrate that our bounds are sometimes better than known ones. For outerplanar graphs without internal triangles, we determine the extremal graphs with the maximum and minimum signless Laplacian spectral radii.

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs.

In this paper, we discuss the conditions under which blow-up occurs for the solutions of reaction-diffusion equations on networks. The analysis of this class of problems includes the existence of blow-up in finite time and the determination of the blow-up time and the corresponding blow-up rate. In addition, when the solution blows up, we give estimates for the blow-up time and also provide the blow-up rate. Finally, we show some numerical illustrations which describe the main results.