We define four minor-closed classes of signed graphs in terms of embeddability in the annulus, projective plane, torus, and Klein bottle. We give the full list of 20 excluded minors for the smallest class and make a conjecture about the largest class.

The diffusion-controlled reaction kA + M is known IO be strongly dependent on Ruchlations in dimensions d < d. = 2 / ( k 1). We develop a field-theoretic renormalization group approach to this system which allows explicit calculation of the observables as expansions in c I / ( ~ I ) , where f = d, d . For the density it is found that, asymptotically, n Akr-d/Z. The decay exponent is emcl to all...

where t > 0, E > 0 is a small parameter, and A is a nonnegative self-adjoint (not necessarily bounded) operator in H, converge, as E + 0, to the solution of (1.2). While we borrow Kisynski’s idea of using the functional calculus of the operator A in order to construct a solution of (1.3), our approach is different from Kisynski’s in that we do not employ the techniques of the theory of semigrou...

Cayley-Dickson construction produces a sequence of normed algebras over real numbers. Its consequent applications result in complex numbers , quaternions, octonions, etc. In this paper we formalize the construction and prove its basic properties. The notation and terminology used here have been introduced in the following We use the following convention: u, v, x, y, z, X, Y are sets and r, s ar...

In this paper, we perform the initial and comprehensive study on the problem of measuring node relevance on signed social networks. We design numerous relevance measurements for signed social networks from both local and global perspectives and investigate the connection between signed relevance measurements, balance theory and signed network properties. Experimental results are conducted to st...

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G,σ) (S = (G,μ)) where G = (V, E) is a graph called underlying graph of S and σ : E → (e1, e2, ..., ek) (μ : V → (e1, e2, ..., ek)) is a function, where each ei ∈ {+,−}. Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviated a signed graph or a marked graph. Given a ...

Let D be a finite and simple digraph with the vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If∑ x∈N[v] f(x) ≥ 1 for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed dominating function on D. The sum f(V (D)) is called the weight w(f) of f . The minimum of weights w(f), taken over all signed dominating function...

In this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized.