Degree Conditions and Degree Bounded Trees

In this paper, we give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and $A\subseteq V(G)$. We denote by ${\sigma }_k(A)$ the minimum value of the degree sum in G of any k pairwise nonadjacent vertices of A, and by $w(G-A)$ the number of components of the subgraph $G-A$ of G induced by $V(G)-A$. Our main results are the following: (i) If ${\sigma }_k(A)⩾|G|-1$, then G contains a tree T with maximum degree ⩽k and $A\subseteq V(T)$. (ii) If ${\sigma }_{k-w(G-A)}(A)⩾|A|-1$, then G contains a spanning tree T with $d_T(x)⩽k$ for any $x\in A$. These are generalizations of the result by S. Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Seminar Univ. Humburg 43 (1975) 263–267] and degree conditions are sharp.     

Rank,term rank and chromatic number of a graph

Let G be a graph with a nonempty edge set, we denote the rank of the adjacency matrix of G and the term rank of G, by $\mathrm{rk}(G)$ and $\mathrm{Rk}(G)$, respectively. It was conjectured [C. van Nuffelen, Amer. Math. Monthly 83 (1976) 265–266], for any graph G, $\chi (G)?\mathrm{rk}(G)$. The first counterexample to this conjecture was obtained by Alon and Seymour [J. Graph Theor. 13 (1989) 523–525]. Recently, Fishkind and Kotlov [Discrete Math. 250 (2002) 253–257] have proved that for any graph G, $\chi (G)?\mathrm{Rk}(G)$. In this Note we improve Fishkind–Kotlov upper bound and show that $\chi (G)?\frac{\mathrm{rk}(G)+\mathrm{Rk}(G)}{2}$. To cite this article: S. Akbari, H.-R. Fanaï, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T. Let w be an element of the Weyl group W of G, and let $Z(\stackrel{?}{w})$ be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression $\stackrel{?}{w}$ of w with respect to the data $(G,B,T)$.In this article we give complete characterization of the expressions $\stackrel{?}{w}$ such that the corresponding BSDH variety $Z(\stackrel{?}{w})$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.     

Parabolic subgroups and automorphism groups of Schubert varieties

Let G be a simple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers, B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and $X(w)$ be the Schubert variety in $G/B$ corresponding to w. In this article we show that given any parabolic subgroup P of G containing B properly, there is an element $w\in W$ such that P is the connected component, containing the identity element of the group of all algebraic automorphisms of $X(w)$.     

A new method for lower bounds of L-functions

Let $L(s\text{,}\pi \text{,}r)$ be an L-function which appears in the Langlands–Shahidi theory. We give a lower bound for $L(s\text{,}\pi \text{,}r)$ when $\mathfrak{R}(s)=1$ using Eisenstein series. This method is applicable even when $L(s\text{,}\pi \text{,}r)$ is not known to be absolutely convergent for $\mathfrak{R}(s)>1$. To cite this article: S.S. Gelbart et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Histories in Path Graphs

For a given graph G and a positive integer r the r-path graph, $P_r(G)$, has for vertices the set of all paths of length r in G. Two vertices are adjacent when the intersection of the corresponding paths forms a path of length $r-1$, and their union forms either a cycle or a path of length $r+1$ in G. Let $P_r^k(G)$ be the k-iteration of r-path graph operator on a connected graph G. Let H be a subgraph of $P_r^k(G)$. The k-history $P_r^{-k}(H)$ is a subgraph of G that is induced by all edges that take part in the recursive definition of H. We present some general properties of k-histories and give a complete characterization of graphs that are k-histories of vertices of 2-path graph operator.