Diagonally convex directed polyominoes and even trees: a bijection and related issues

We present a simple bijection between diagonally convex directed (DCD) polyominoes with n diagonals and plane trees with 2n edges in which every vertex has even degree (even trees), which specializes to a bijection between parallelogram polyominoes and full binary trees. Next we consider a natural definition of symmetry for DCD-polyominoes, even trees, ternary trees, and non-crossing trees, and show that the number of symmetric objects of a given size is the same in all four cases.     

The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.     

Bijections and symmetries for the factorizations of the long cycle

We study the factorizations of the permutation (1,2,…,n)$(1,2,\dots ,n)$ into k factors of given cycle types. Using representation theory, Jackson obtained for each k   an elegant formula for counting these factorizations according to the number of cycles of each factor. In the cases k=2,3$k=2,3$ Schaeffer and Vassilieva gave a combinatorial proof of Jackson?s formula, and Morales and Vassilieva obtained more refined formulas exhibiting a surprising symmetry property. These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of the combinatorial properties of the factorizations of (1,2,…,n)$(1,2,\dots ,n)$ into k factors for all k  . We thereby obtain refinements of Jackson?s formulas which extend the cases k=2,3$k=2,3$ treated by Morales and Vassilieva. Our bijections are described in terms of “constellations”, which are graphs embedded in surfaces encoding the transitive factorizations of permutations.     

The butterfly decomposition of plane trees

We introduce the notion of doubly rooted plane trees and give a decomposition of these trees, called the butterfly decomposition, which turns out to have many applications. From the butterfly decomposition we obtain a one-to-one correspondence between doubly rooted plane trees and free Dyck paths, which implies a simple derivation of a relation between the Catalan numbers and the central binomial coefficients. We also establish a one-to-one correspondence between leaf-colored doubly rooted plane trees and free Schröder paths. The classical Chung-Feller theorem as well as some generalizations and variations follow quickly from the butterfly decomposition. We next obtain two involutions on free Dyck paths and free Schröder paths, leading to parity results and combinatorial identities. We also use the butterfly decomposition to give a combinatorial treatment of Klazar's generating function for the number of chains in plane trees. Finally we study the total size of chains in plane trees with n edges and show that the average size of such chains tends asymptotically to (n+9)/6.     

Ordering trees by the Laplacian coefficients

Motivated by a Mohar’s paper proposing “how to order trees by the Laplacian coefficients”, we investigate a partial ordering of trees with diameters 3 and 4 by the Laplacian coefficients. These results are used to determine several orderings of trees by the Laplacian coefficients.     

Counting colorful multi-dimensional trees

LetV be a disjoint union ofr finite setsV ₁,...,V _r (colors). A collectionT of subsets ofV iscolorful if each member ifT contains at most one point of each color. Ak-dimensional colorful tree is a colorful collectionT of subsets ofV, each of sizek+1, such that if we add toT all the colorful subsets ofV of sizek or less, we get aQ-acyclic simplicial complex _T We count (using the Binet-Cauchy theorem) thek-dimensional colorful trees onV (for allk), where each treeT is counted with weight . The result confirms, in a way, a formula suggested by Bolker. (fork-r–1). It extends, on one hand, a result of Kalai on weighted counting ofk-dimensional trees and, on the other hand, enumeration formulas for multi-partite (1-dimensional) trees. All these results are extensions of Cayley's celebrated treecounting formula, now 100 years old.     

The Laplacian spectral radii of trees with degree sequences

In this paper, we characterize all extremal trees with the largest Laplacian spectral radius in the set of all trees with a given degree sequence. Consequently, we also obtain all extremal trees with the largest Laplacian spectral radius in the sets of all trees of order n with the largest degree, the leaves number and the matching number, respectively.     

New upper bounds on the spectral radius of trees with the given number of vertices and maximum degree

This paper studies the problem of estimating the spectral radius of trees with the given number of vertices and maximum degree. We obtain the new upper bounds on the spectral radius of the trees, and the results are the best upper bounds expressed by the number of vertices and maximum degree, at present.     

The least eigenvalue of the complements of trees

Let be the set of the complements of trees of order n. In this paper, we characterize the unique graph whose least eigenvalue attains the minimum among all graphs in .     

On the nullity of graphs with pendant trees

The nullity of a graph is defined as the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph. We investigate a class of graphs with pendant trees, and express the nullity of such graph in terms of that of its subgraphs. As an application of our results, we characterize unicyclic graphs with a given nullity.     

The spectra of some trees and bounds for the largest eigenvalue of any tree

Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let n_k−j+1 and d_k−j+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (n_k = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d₁, d₂, …, d_k−1, d_k ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.     

Ordering of trees with fixed matching number by the Laplacian coefficients

Let G be a simple undirected graph with the characteristic polynomial of its Laplacian matrix L(G), . Aleksandar Ili? [A. Ili?, Trees with minimal Laplacian coefficients, Comput. Math. Appl. 59 (2010) 2776-2783] identified n-vertex trees with given matching number q which simultaneously minimize all Laplacian coefficients. In this paper, we give another proof of this result. Generalizing the approach in the above paper, we determine n-vertex trees with given matching number q which have the second minimal Laplacian coefficients. We also identify the n-vertex trees with a perfect matching having the largest and the second largest Laplacian coefficients, respectively. Extremal values on some indices, such as Wiener index, modified hyper-Wiener index, Laplacian-like energy, incidence energy, of n-vertex trees with matching number q are obtained in this paper.     

Distance spectral radius of trees with fixed number of pendent vertices

The distance spectral radius ρ(G)$\rho (G)$ of a graph G   is the largest eigenvalue of the distance matrix D(G)$D(G)$. In this paper, we characterize the graph with minimum distance spectral radius among trees with fixed number of pendent vertices.     

An explicit formula for eigenvalues of Bethe trees and upper bounds on the largest eigenvalue of any tree

A Bethe tree B_d,k is a rooted unweighted of k levels in which the root vertex has degree equal to d, the vertices at level j(2?j?k-1) have degree equal to (d+1) and the vertices at level k are the pendant vertices. In this paper, we first derive an explicit formula for the eigenvalues of the adjacency matrix of B_d,k. Moreover, we give the corresponding multiplicities. Next, we derive an explicit formula for the simple nonzero eigenvalues, among them the largest eigenvalue, of the Laplacian matrix of B_d,k. Finally, we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any tree T. These upper bounds are given in terms of the largest vertex degree and the radius of T, and they are attained if and only if T is a Bethe tree.     

Extremal problems on detectable colorings of trees

Let G be a connected graph of order 3 or more and let be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G, the color code of v is the k-tuple c(v)=(a₁,a₂,…,a_k), where a_i is the number of edges incident with v that are colored i (1?i?k). The coloring c is called detectable if distinct vertices have distinct color codes; while the detection number det(G) of G is the minimum positive integer k for which G has a detectable k-coloring. For each integer n?3, let D_T(n) be the maximum detection number among all trees of order n and d_T(n) the minimum detection number among all trees of order n. The numbers D_T(n) and d_T(n) are determined for all integers n?3. Furthermore, it is shown that for integers k?2 and n?3, there exists a tree T of order n having det(T)=k if and only if d_T(n)?k?D_T(n).     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

On a combinatorial problem related to permanents

Some numerical results will be presented concerning the largest and smallest values of square permanents containing the integers 1, 2, 3, ...,n ² n=1(1)8.Dedicated to Peter Naur on the occasion of his 60th birthday     

Inverses of trees

LetT be a tree with a perfect matching. It is known that in this case the adjacency matrixA ofT is invertible and thatA ^?1 is a (0, 1, ?1)-matrix. We show that in factA ^?1 is diagonally similar to a (0, 1)-matrix, hence to the adjacency matrix of a graph. We use this to provide sharp bounds on the least positive eigenvalue ofA and some general information concerning the behaviour of this eigenvalue. Some open problems raised by this work and connections with Möbius inversion on partially ordered sets are also discussed.