Related papers: Quadratic starlike trees
The index of a graph is the largest eigenvalue of its adjacency matrix. A starlike is a tree having a unique vertex of degree $r>2$. We show how to order the starlike trees with $n>3$ by their indices. In particular, the indices of starlike…
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…
We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs…
In this paper we introduce the quadratic Jaco graph. The characteristics, properties and some graph invariants of quadratic Jaco graphs are discussed. The observation that quadratic Jaco graphs are well-defined in respect of complete graphs…
The family of trees with palindromic characteristic polynomials is characterized. Large families of graphs with this property are found as well.
We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from…
We examine the behavior of the sequences of $p$-adic valuations of quadratic polynomials with integer coefficients for an odd prime $p$ through tree representations. Under this representation, a finite tree corresponds to a periodic…
We compute the quadratic embedding constant for complete bipartite graphs with disjoint edges removed. Moreover, we study the quadratic embedding property for theta graphs, i.e., graphs consisting of three paths with common initial points…
The transmission of a vertex in a connected graph is the sum of its distances to all the other vertices. A graph is transmission irregular (TI) when all of its vertices have mutually distinct transmissions. In an earlier paper, Al-Yakoob…
A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Recently, Csikvari proved the existence of integral trees of any even diameter. In the odd case, integral trees have been constructed with…
A graph $G$ is \textit{asymmetric} if its automorphism group of vertices is trivial. Asymmetric graphs were introduced by Erd\H{o}s and R\'{e}nyi in 1963. They showed that the probability of a graph on $n$ vertices being asymmetric tends to…
We characterize positive definiteness for some family of matrices. As an application we derive explicit value of the quadratic embedding constants of the path graphs.
We introduce classes of edge-colourings of the complete graph -- that we call nice and beautiful -- and study how many heterochromatic spanning trees appear under such colourings. We prove that if the colouring is nice, there is at least a…
A squaregraph is a plane graph in which each internal face is a $4$-cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the semi-strong product of an outerplanar…
Graph eigenvalues are examples of totally real algebraic integers, i.e. roots of real-rooted monic polynomials with integer coefficients. Conversely, the fact that every totally real algebraic integer occurs as an eigenvalue of some finite…
This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of…
The conception of multi-alphabetical genetics is represented. Matrix forms of the representation of the multi-level system of molecular-genetic alphabets have revealed algebraic properties of this system. These properties are connected with…
We describe all the trees with the property that the corresponding edge ideal of the square of the tree has a linear resolution. As a consequence, we give a complete characterization of those trees $T$ for which the square is co-chordal,…
We describe our current understanding on the phase transition phenomenon of the graph Laplacian eigenvectors constructed on a certain type of unweighted trees, which we previously observed through our numerical experiments. The eigenvalue…
A Hypercube $Q_n$ is a graph in which the vertices are all binary vectors of length n, and two vertices are adjacent if and only if their components differ in exactly one place. A galaxy or a star forest is a union of vertex disjoint stars.…