We introduce three new cut tree structures of graphs G in which the vertex set of the tree is a partition of V (G) and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup [8] presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existen...

In this paper, we study the signless Laplacian spectral radius of bicyclic graphs with given number of pendant vertices and characterize the extremal graphs.

‎A tree containing exactly two non-pendant vertices is called a double-star‎. ‎Let $k_1$ and $k_2$ be two positive integers‎. ‎The double-star with degree sequence $(k_1+1‎, ‎k_2+1‎, ‎1‎, ‎ldots‎, ‎1)$ is denoted by $S_{k_1‎, ‎k_2}$‎. ‎It is known that a cubic graph has an $S_{1,1}$-decomposition if and only if it contains a perfect matching‎. ‎In this paper‎, ‎we study the $S_{1,2}$-decomposit...

*Correspondence: [email protected] 1School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, Jiangsu 224002, P.R. China Full list of author information is available at the end of the article Abstract A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013) and Liu et al. (Electron. J. Linear ...

The status of a vertex v in connected graph G is defined as the sum distances from to all other vertices . minimum We give smallest and largest values for tree with fixed parameters such diameter, number pendant vertices, odd degree two, characterize unique extremal trees.

One theorem of Nemhauser and Trotter [10] ensures that, under certain conditions, a stable set of a graph G can be enlarged to a maximum stable set of this graph. For example, any stable set consisting of only simplicial vertices is contained in a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for trees of order greater than one, where, in fact, all the ...

Mathematical descriptors of molecular structure, such as various topological indices, have been widely used in structure-property-activity studies (see [5, 6, 12]). Among the numerous topological indices considered in chemical graph theory, only a few have been found noteworthy in practical application (see [10]). One of these is the connectivity index or Randić index. The Randić index of an or...

A connected graph is called a c-cyclic graph if it contains n vertices and n + c − 1 edges. Let C(n, k, c) denote the class of connected c-cyclic graphs with n vertices and k pendant vertices. Recently, the unique extremal graph, which has greatest (respectively, signless) Laplacian spectral radius, in C(n, k, c) has been determined for 0 ≤ c ≤ 3, k ≥ 1 and n ≥ 2c + k + 1. In this paper, the un...