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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.
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
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}
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 $k$-ve