Null Decomposition of Trees

Let $T$ be a tree, we show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, and use it in order to get formulas for independence number and matching number of a tree. We also prove that the number of maximum matchings in a tree is related to the null decomposition.Comment: 23 pages, 6 figure

Maximum and minimum nullity of a tree degree sequence

The nullity of a graph is the multiplicity of the eigenvalue zero in its adjacency spectrum. In this paper, we give a closed formula for the minimum and maximum nullity among trees with the same degree sequence, using the notion of matching number and annihilation number. Algorithms for constructing such minimum-nullity and maximum-nullity trees are described.Comment: 13 pages and 3 figure

On the structure of the fundamental subspaces of acyclic matrices with 00 in the diagonal

A matrix is called acyclic if replacing the diagonal entries with $0$, and the nonzero diagonal entries with $1$, yields the adjacency matrix of a forest. In this paper we show that null space and the rank of a acyclic matrix with $0$ in the diagonal is obtained from the null space and the rank of the adjacency matrix of the forest by multipliying by non-singular diagonal matrices. We combine these methods with an algorithm for finding a sparsest basis of the null space of a forest to provide an optimal time algorithm for finding a sparsest basis of the null space of acyclic matrices with $0$ in the diagonal.Comment: 7 page

S-trees

In this paper two new graph operations are introduced, and with them the S-trees are studied in depth. This allows to find $\{-1,0,1\}$-basis for all the fundamental subspaces of the adjacency matrix of any tree, and to understand in detail the matching structure of any tree.Comment: 26 pages, 9 figure

On the null structure of bipartite graphs without cycles of length a multiple of 4

In this work we study the null space of bipartite graphs without cycles of length multiple of $4$, and its relation to structural properties. We decompose them into two subgraphs: $C_N(G)$ and $C_S(G)$. $C_N(G)$ has perfect matching and its adjacency matrix is nonsingular. $C_S(G)$ has a unique maximum independent set and the dimension of its null space equals the dimension of the null space of $G$. Even more, we show that the fundamental spaces of $G$ are the direct sum of the fundamental spaces of $C_N(G)$ and $C_S(G)$. We also obtain formulas relating the independence number and the matching number of a $C_{4k}$-free bipartite graph with $C_N(G)$ and $C_S(G)$, and the dimensions of the fundamental spaces. Among other results, we show that the rank of a $C_{4k}$-free bipartite graph is twice its matching number, generalizing a result for trees due to Bevis et al \cite{bevis1995ranks}, and Cvetkovi\'c and Gutman \cite{D1972}. About maximum independent sets, we show that the intersection of all maximum independent sets of a $C_{4k}$-free bipartite graph coincides with the support of its null space.Comment: 20 pages, 3 figure

Walks on Unitary Cayley Graphs and Applications

In this paper, we determine an explicit formula for the number of walks in $X_n = \textsf{Cay}(\mathbb{Z}_n,\mathbb{U}_n)$, the unitary Cayley Graphs of order $n$, between any pair of its vertices. With this result, we give the number of representations of a fixed residue class $\bmod{}n$ as the sum of $k$ units of $\mathbb{Z}_n$

A {−1,0,1}\{-1,0,1\}- and sparsest basis for the null space of a forest in optimal time

Given a matrix, the Null Space Problem asks for a basis of its null space having the fewest nonzeros. This problem is known to be NP-complete and even hard to approximate. The null space of a forest is the null space of its adjacency matrix. Sander and Sander (2005) and Akbari et al. (2006), independently, proved that the null space of each forest admits a $\{-1,0,1\}$-basis. We devise an algorithm for determining a sparsest basis of the null space of any given forest which, in addition, is a $\{-1,0,1\}$-basis. Our algorithm is time-optimal in the sense that it takes time at most proportional to the number of nonzeros in any sparsest basis of the null space of the input forest. Moreover, we show that, given a forest $F$ on $n$ vertices, the set of those vertices $x$ for which there is a vector in the null space of $F$ that is nonzero at $x$ and the number of nonzeros in any sparsest basis of the null space of $F$ can be found in $O(n)$ time.Comment: 9 page

2-switch transition on unicyclic graphs and pseudoforest

In the present work we prove that given any two unicycle graphs (pseudoforests) that share the same degree sequence there is a finite sequence of 2-switches transforming one into the other such that all the graphs in the sequence are also unicyclic graphs (pseudoforests).Comment: 12 pages. arXiv admin note: text overlap with arXiv:2004.1116

2-switch: transition and stability on graphs and forests

Given any two forests with the same degree sequence, we show in an algorithmic way that one can be transformed into the other by a sequence of 2-switches in such a way that all the intermediate graphs of the transformation are forests. We also prove that the 2-switch operation perturbs minimally some well-known integer parameters in families of graphs with the same degree sequence. Then, we apply these results to conclude that the studied parameters have the interval property on those families.Comment: 23 pages, 2 figure

Using digrahs to compute determinat, permanent and Drazin (group) inverse of circulant matrices with two parameters

In this work we present closed formulas for the determinant, the permanent, the inverse and the Drazin inverse of circulant matrices with two non-zero coefficients