On a graph property generalizing planarity and flatness

We introduce a topological graph parameter σ(G), defined for any graph G. This parameter characterizes subgraphs of paths, outerplanar graphs, planar graphs, and graphs that have a flat embedding as those graphs G with σ(G)≤1,2,3, and 4, respectively. Among several other theorems, we show that if H is a minor of G, then σ(H)≤σ(G), that σ(K _n )=n−1, and that if H is the suspension of G, then σ(H)=σ(G)+1. Furthermore, we show that μ(G)≤σ(G) + 2 for each graph G. Here μ(G) is the graph parameter introduced by Colin de Verdière in [2].     

On a property of solutions to the poisson equation on polygons

It is proved that an equilateral triangle is the unique polygon on which the solution of the equation Δu=1 with homogeneous boundary conditions is an algebraic polynomial, and moreover, the degree of this polynomial is equal to 3. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 178–180, August, 1999.     

On the solvability of a boundary value problem for a fourth-order equation on a graph

We consider a boundary value problem for a fourth-order equation on a graph modeling elastic deformations of a plane rod system with conditions of rigid connection at the vertices. Conditions for the unique solvability are stated. We also present sufficient conditions for the problem to be degenerate.     

On a decay property of solutions to the Haraux-Weissler equation

We give a sufficient condition that non-radial H¹-solutions to the Haraux-Weissler equation should belong to the weighted Sobolev space , where ρ is the weight function exp(|x|²/4). Our result provides, in some sense, a connection between the solutions obtained by ODE method and those by variational approach in the space .     

On an inverse type problem for the heat equation in parabolic regular graph domains

In this paper we prove some results concerning inverse/free boundary type problems, below the continuous threshold, for the heat equation in the setting of parabolic regular graph domains.     

On an isomorphism problem on the closed-set lattice of a graph

Let (G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism : V(G)(G) induces a bijection : V(G)V(G) such that for each x in V(G), (x)=x' in V(G') iff ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism : (G)(G), the bijection induced by is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C ₄-free graphs and all covering graphs of finite lattices are strongly sensitive.     

On a functional equation related to the Matsumoto—Yor property

Summary. A functional equation arising from the independence properties of some transformations of independent generalized inverse Gaussian and gamma variables is completely solved under the local integrability assumption.     

Reduction method for a fourth-order equation on a graph

We justify a method that permits one to reduce a boundary value problem on a graph to a problem on a narrower subset provided that the right-hand side of the differential equation is identically zero on some subgraph of the original graph. We find the signs of the coefficients in the boundary conditions of the reduced problem and clarify the relationship between these coefficients.     

On a class of packet-switching networks on an infinite graph

A class of open queueing networks with packet switching is discussed. The configuration graph of the network may be finite or infinite. The external messages are divided into standard pieces (packets) each of which is transmitted as a single message. The messages are addressed, as a rule, to nearest neighbours and thereby the network may be treated as a small perturbation of the collection of isolated servers. The switching rule adopted admits overtaking: packets which appeared later may reach the delivery node earlier. The transmission of a message is completed when its last packet reaches the destination node. The main result of this paper is that the network possesses a unique stationary working regime. Weak dependence properties of this regime are established as well as the continuity with respect to the parameters of the external message flows.     

On the hamiltonian path graph of a graph

The hamiltonian path graph H(F) of a graph F is that graph having the same vertex set as F and in which two vertices u and v are adjacent if and only if F contains a hamiltonian u ? v path. First, in response to a conjecture of Chartrand, Kapoor and Nordhaus, a characterization of nonhamiltonian graphs isomorphic to their hamiltonian path graphs is presented. Next, the maximum size of a hamiltonian graph F of given order such that K?_d ? H(F) is determined. Finally, it is shown that if the degree sum of the endvertices of a hamiltonian path in a graph F with at least five vertices is at least |V(F)| + t(t ? 0), then H(F) contains a complete subgraph of order t + 4.     

The Hicks property for a variational problem on a graph

In this paper we use the Hicks property for a variational problem on a graph. For an elastic system defined on a graph we state a well-posed problem which implies the definition and the study of the influence function.     

On the spectrum of an infinite graph

We exhibit an example of an infinite locally finite graph such that its adjacency operator is not self-adjoint. This gives a negative answer to a conjecture of Mohar [1].     

A surprising property of the least eigenvalue of a graph

Let λ(G) be the least eigenvalue of a graph G. A real number r has the induced subgraph property provided λ(G)<r implies G has an induced subgraph H with λ(H)=r. It is shown that the only numbers with the induced subgraph property are 0, ?1, ?$\text{2}$, and ?2.