On a signless Laplacian spectral characterization of -shape trees

Let M be an associated matrix of a graph G (the adjacency, Laplacian and signless Laplacian matrix). Two graphs are said to be cospectral with respect to M if they have the same M spectrum. A graph is said to be determined by M spectrum if there is no other non-isomorphic graph with the same spectrum with respect to M. It is shown that T-shape trees are determined by their Laplacian spectra. Moreover among them those are determined by their adjacency spectra are characterized. In this paper, we identify graphs which are cospectral to a given T-shape tree with respect to the signless Laplacian matrix. Subsequently, T-shape trees which are determined by their signless Laplacian spectra are identified.     

Signless Laplacian spectral characterization of line graphs of T-shape trees

A T-shape tree is a tree with exactly one vertex of maximum degree 3. The line graphs of the T-shape trees are triangles with a hanging path at each vertex. Let C_a,b,c be such a graph, where a, b and c are the lengths of the paths. In this paper, we show that line graphs of T-shape trees, with the sole exception of C_a,a,2a+1, are determined by the spectra of their signless Laplacian matrices. For the graph C_a,a,2a+1 we identify the unique non-isomorphic graph sharing the same signless Laplacian characteristic polynomial.     

Tree Sets

We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and matroids etc.Unlike graph-theoretical or order trees, these tree sets can provide a suitable formalization of tree structure also for infinite graphs, matroids, and set partitions. Order trees reappear as oriented tree sets.We show how each of the above structures defines a tree set, and which additional information, if any, is needed to reconstruct it from this tree set.     

Phase Transition of Mixed Type <Emphasis Type="Italic">p</Emphasis>-Adic λ-Ising Model on Cayley Tree

In the present paper, we consider an interaction of the nearest-neighbors and next nearest-neighbors for the mixed type p-adic λ-Ising model with spin values {?1, +1} on the Cayley tree of order two.We obtained the uniqueness and existence of the p-adic quasi Gibbs measures for the model. Thereafter, as a main result, we proved the occurrence of phase transition for the p-adic λ-Ising model on the Cayley tree of order two. To establish the results, we employed some properties of p-adic numbers. Therefore, our results are not valid in the real case.     

On Models of Directed Path Graphs Non Rooted Directed Path Graphs

A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted directed path graph is the intersection graph of a family of directed subpaths of a rooted tree. Clearly, rooted directed path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to find such characterizations for rooted directed path graphs. With the purpose of proving knowledge in this direction, we show in this paper properties of directed path models that can not be rooted for chordal graphs with any leafage and with leafage four. Therefore, we prove that for leafage four directed path graphs minimally non rooted directed path graphs have a unique asteroidal quadruple, and can be characterized by the presence of certain type of asteroidal quadruples.     

On the linear extension complexity of stable set polytopes for perfect graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph $G$ depending linearly on the size of $G$ and involving the depth of a decomposition tree of $G$ in terms of basic perfect graphs.     

Tree-Like Structure Graphs with Full Diversity of Balls

Under study is the diversity of metric balls in connected finite ordinary graphs considered as a metric space with the usual shortest-path metric. We investigate the structure of graphs in which all balls of fixed radius i are distinct for each i less than the diameter of the graph. Let us refer to such graphs as graphs with full diversity of balls. For these graphs, we establish some properties connected with the existence of bottlenecks and find out the configuration of blocks in the graph. Using the obtained properties, we describe the tree-like structure graphs with full diversity of balls.     

Note the T-shape tree is determined by its Laplacian spectrum

A T-shape is a tree with exactly one of its vertices having maximal degree 3. It is proved in this paper that the T-shape tree is determined by its Laplacian spectrum.