Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuadratic starlike trees
116
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we introduce the notion of the quadratic graph, that is a graph whose eigenvalues are integral or quadratic algebraic integral, and determine nine infinite families of quadratic starlike trees, which are just all the quadratic starlike trees including integral starlike trees. Thus the quadratic starlike trees are completely characterized, and moreover, the display expressions for the characteristic polynomials of the quadratic starlike trees are also given.
rate research
Read More
In this short note, we first present a simple bijection between binary trees and colored ternary trees and then derive a new identity related to generalized Catalan numbers.
A new 2-parameter family of central structures in trees, called central forests, is introduced. Miniekas $m$-center problem and McMorriss and Reids central-$k$-tree can be seen as special cases of central forests in trees. A central forest is defined as a forest $F$ of $m$ subtrees of a tree $T$, where each subtree has $k$ nodes, which minimizes the maximum distance between nodes not in $F$ and those in $F$. An $O(n(m+k))$ algorithm to construct such a central forest in trees is presented, where $n$ is the number of nodes in the tree. The algorithm either returns with a central forest, or with the largest $k$ for which a central forest of $m$ subtrees is possible. Some of the elementary properties of central forests are also studied.
Let $T_{n}$ be the set of rooted labeled trees on $set{0,...,n}$. A maximal decreasing subtree of a rooted labeled tree is defined by the maximal subtree from the root with all edges being decreasing. In this paper, we study a new refinement $T_{n,k}$ of $T_n$, which is the set of rooted labeled trees whose maximal decreasing subtree has $k+1$ vertices.
A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique.
For a connected graph, a {em minimum vertex separator} is a minimum set of vertices whose removal creates at least two connected components. The vertex connectivity of the graph refers to the size of the minimum vertex separator and a graph is $k$-vertex connected if its vertex connectivity is $k$, $kgeq 1$. Given a $k$-vertex connected graph $G$, the combinatorial problem {em vertex connectivity augmentation} asks for a minimum number of edges whose augmentation to $G$ makes the resulting graph $(k+1)$-vertex connected. In this paper, we initiate the study of $r$-vertex connectivity augmentation whose objective is to find a $(k+r)$-vertex connected graph by augmenting a minimum number of edges to a $k$-vertex connected graph, $r geq 1$. We shall investigate this question for the special case when $G$ is a tree and $r=2$. In particular, we present a polynomial-time algorithm to find a minimum set of edges whose augmentation to a tree makes it 3-vertex connected. Using lower bound arguments, we show that any tri-vertex connectivity augmentation of trees requires at least $lceil frac {2l_1+l_2}{2} rceil$ edges, where $l_1$ and $l_2$ denote the number of degree one vertices and degree two vertices, respectively. Further, we establish that our algorithm indeed augments this number, thus yielding an optimum algorithm.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
