A Construction of Cospectral Graphs

In this report, we investigated two questions in the field of spectral graph theory. The first question was whether it is possible to extend recent results to find a large class of graphs uniquely determined by the spectrum of its adjacency matrix. Our investigation led to the discovery of a pair of cospectral graphs which contradicted the existence of such a class. The second question was whether there exists a construction of cospectral graphs that consists of adding a single edge and vertex to a given pair of cospectral graphs. We discovered that such a construction exists, and generated several pairs of cospectral graphs using this method. Further investigation showed that this construction of cospectral graphs is related to two previously studied constructions

Integral trees of diameter 4

An integral tree is a tree whose adjacency matrix has only integer eigenvalues. While most previous work by other authors has been focused either on the very restricted case of balanced trees or on finding trees with diameter as large as possible, we study integral trees of diameter 4. In particular, we characterize all diameter 4 integral trees of the form T(m1, t1) T(m2, t2). In addition we give elegant parametric descriptions of infinite families of integral trees of the form T(m1, t1) · · · T(mn, tn) for any n > 1. We conjecture that we have found all such trees

Descriptive complexity of controllable graphs

Let $G$ be a graph on $n$ vertices with adjacency matrix $A$, and let $\mathbf{1}$ be the all-ones vector. We call $G$ controllable if the set of vectors $\mathbf{1}, A\mathbf{1}, \dots, A^{n-1}\mathbf{1}$ spans the whole space $\mathbb{R}^n$. We characterize the isomorphism problem of controllable graphs in terms of other combinatorial, geometric and logical problems. We also describe a polynomial time algorithm for graph isomorphism that works for almost all graphs.Comment: 14 page

On graphs with cyclic defect or excess

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called {\it graphs with defect or excess $\epsilon$}, respectively. While {\it Moore graphs} (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n-B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs \emph{graphs with cyclic defect or excess}; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the M\"obius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.Comment: 20 pages, 3 Postscript figure

Descriptive complexity of graph spectra.

Two graphs are cospectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are cospectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic are cospectral, and this is not the case with , nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.OZ was supported by CONACyT-Mexico Grant 384665, SS was supported by EPSRC and The Royal Society

Constructing Cospectral and Comatching Graphs

The matching polynomial is a graph polynomial that does not only have interesting mathematical properties, but also possesses meaningful applications in physics and chemistry. For a simple graph, the matching polynomial enumerates the number of matchings of different sizes in it. Two graphs are comatching if they have the same matching polynomial. Two vertices u, v in a graph G are comatching if G\ u and G\ v are comatching. In 1973, Schwenk proved almost every tree has the same characteristic polynomial with another tree. In this thesis, we extend Schwenk's result to maximal limbs and weighted trees. We also give a construction using 1-vertex extensions for comatching graphs and graphs with an arbitrarily large number of comatching vertices. In addition, we use an alternative definition of matching polynomial for multigraphs to derive new identities for the matching polynomial. These identities are tools used towards our 2-sum construction for comatching vertices and comatching graphs

Mean first-passage time for random walks on generalized deterministic recursive trees

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