................................................................................................................. iii ACKNOWLEDGMENTS............................................................................................ viii LIST OF TABLES.......................................................................................................... x LIST OF FIGURES ..............................

We prove that we can specify by formulas of monadic second-order logic the unique planar embedding of a 3-connected planar graph. If the planar graph is not 3-connected but given with a linear order of its set of edges, we can also define a planar embedding by monadic second-order formulas. We cannot do so in general without the ordering, even for 2-connected planar graphs. The planar embedding...

The KP equation is a nonlinear dispersive wave which provides an excellent model for resonant interactions of shallow-water waves. It well known that regular soliton solutions the may be constructed from points in totally nonnegative Grassmannian Gr(N,M)?0. Kodama and Williams studied asymptotic patterns (tropical limit) solitons, called graphs, showed they correspond to Postnikov's Le-diagrams...

This article establishes that the split decomposition of graphs introduced by Cunnigham, is definable in Monadic Second-Order Logic.This result is actually an instance of a more general result covering canonical graph decompositions like the modular decomposition and the Tutte decomposition of 2-connected graphs into 3-connected components. As an application, we prove that the set of graphs hav...

Large knowledge graphs increasingly add great value to various applications that require machines to recognize and understand queries and their semantics, as in search or question answering systems. These applications include Google search, Bing search, IBM’s Watson, but also smart mobile assistants as Apple’s Siri, Google Now or Microsoft’s Cortana. Popular knowledge graphs like DBpedia, YAGO ...

cospectrality of two graphs measures the differences between the ordered spectrum of these graphs in various ways. actually,the origin of this concept came back to richard brualdi's problems that are proposed in cite{braldi}:let $g_n$ and $g'_n$ be two nonisomorphic simple graphs on $n$ vertices with spectra$$lambda_1 geq lambda_2 geq cdots geq lambda_n ;;;text{and};;; lambda'_1 geq lambda'_2 g...