arXiv subjects

Irene Sciriha

Publications and source records attributed to Irene Sciriha.

9 recordsLinked to original sources

arXiv2026

A Note on the Laplacian Eigenvectors of Threshold Graphs

Threshold graphs are graphs that can be characterized in a number of different ways. For example, they are graphs that are $P_4,\ C_4,\ 2K_2$--free. They may also be characterized by a finite sequence of positive integers $a_1, \ldots, a_r$, such that $a_1\geqslant 2$ and $a_1 + a_2 + \cdots + a_r = |V(G)|$. Threshold graphs have the remarkable property that all graphs of the same order share a common integer Laplacian eigenbasis. This property characterizes threshold graphs. This result was proved in \cite{MachareteDelVecchio}. We give a different proof of the same result.

math.CO↗

arXiv2020

On singular signed graphs with nullspace spanned by a full vector: Signed nut graphs

A signed graph has edge weights drawn from the set $\{+1,-1\}$, and is termed sign-balanced if it is equivalent to an unsigned graph under the operation of sign switching; otherwise it is called sign-unbalanced. A nut graph has a one dimensional kernel with a corresponding eigenvector that is full. In this paper we generalise the notion of nut graphs to signed graphs. Orders for which unsigned regular nut graphs exist were determined recently for the degrees up to $11$. By extending the definition to signed nut graphs, we find all pairs $(ρ, n)$ for which a $ρ$-regular nut graph (sign-balanced or sign-unbalanced) of order $n$ exists with $ρ\le 11$. We devise a construction for signed nut graphs based on a smaller `seed' graph, giving infinite series of both sign-balanced and sign-unbalanced $ρ$-regular nut graphs. All orders for which a complete sign-unbalanced nut graph exists are characterised; they have underlying graph $K_n$ with $n \equiv 1 \pmod 4$. All orders for which a regular sign-unbalanced nut graph with $ρ= n - 2$ exists are also characterised; they have an underlying cocktail-party graph $\mathrm{CP}(n)$ with even $n \geq 8$.

math.CO↗

arXiv2020

Nullspace Vertex Partition in Graphs

The core vertex set of a graph is an invariant of the graph. It consists of those vertices associated with the non-zero entries of the nullspace vectors of a $\{0,1\}$-adjacency matrix. The remaining vertices of the graph form the core--forbidden vertex set. For graphs with independent core vertices, such as bipartite minimal configurations and trees, the nullspace induces a well defined three part vertex partition. The parts of this partition are the core vertex set, their neighbours and the remote core--forbidden vertices. The set of the remote core--forbidden vertices are those not adjacent to any core vertex. We show that this set can be removed, leaving the nullity unchanged. We show that for graphs with independent core vertices, the submatrix of the adjacency matrix defining the edges incident to the core vertices determines the nullity of adjacency matrix. To maximize the number of edges for optimal network graphs with a specified nullity, we determine which perturbations make up sufficient conditions for the core vertex set of the adjacency matrix of a graph to be preserved on adding edges.

math.CO↗

arXiv2020

On the Displacement of Eigenvalues when Removing a Twin Vertex

Twin vertices of a graph have the same open neighbourhood. If they are not adjacent, then they are called duplicates and contribute the eigenvalue zero to the adjacency matrix. Otherwise they are termed co-duplicates, when they contribute $-1$ as an eigenvalue of the adjacency matrix. On removing a twin vertex from a graph, the spectrum of the adjacency matrix does not only lose the eigenvalue $0$ or $-1$. The perturbation sends a rippling effect to the spectrum. The simple eigenvalues are displaced. We obtain a closed formula for the characteristic polynomial of a graph with twin vertices in terms of two polynomials associated with the perturbed graph. These are used to obtain estimates of the displacements in the spectrum caused by the perturbation.

math.SP↗

arXiv2019

Existence of Regular Nut Graphs and the Fowler Construction

In this paper the problem of the existence of regular nut graphs is addressed. A generalization of Fowler's Construction which is a local enlargement applied to a vertex in a graph is introduced to generate nut graphs of higher order. Let $N(ρ)$ denote the set of integers $n$ such that there exists a regular nut graph of degree $ρ$ and order $n$. It is proven that $N(3) = \{12\} \cup \{2k : k \geq 9\}$ and that $N(4) = \{8,10,12\} \cup \{n: n \geq 14\}$. The problem of determining $N(ρ)$ for $ρ> 4$ remains completely open.

math.CO↗

arXiv2019

Existence of regular nut graphs for degree at most 11

A nut graph is a singular graph with one-dimensional kernel and corresponding eigenverctor with no zero elements. The problem of determining the orders $n$ for which $d$-regular nut graphs exist was recently posed by Gauci, Pisanski and Sciriha. These orders are known for $d \leq 4$. Here we solve the problem for all remaining cases $d\leq 11$ and determine the complete lists of all $d$-regular nut graphs of order $n$ for small values of $d$ and $n$. The existence or non-existence of small regular nut graphs is determined by a computer search. The main tool is a construction that produces, for any $d$-regular nut graph of order $n$, another $d$-regular nut graph of order $n + 2d$. If we are given a sufficient number of $d$-regular nut graphs of consecutive orders, called seed graphs, this construction may be applied in such a way that the existence of all $d$-regular nut graphs of higher orders is established. For even $d$ the orders $n$ are indeed consecutive, while for odd $d$ the orders $n$ are consecutive even numbers. Furthermore, necessary conditions for combinations of order and degree for vertex-transitive nut graphs are derived.

math.CO↗

arXiv2019

On the Walks and Bipartite Double Coverings of Graphs with the same Main Eigenspace

The main eigenvalues of a graph $G$ are those eigenvalues of the $(0,1)$-adjacency matrix $\mathbf A$ having a corresponding eigenvector not orthogonal to $\mathbf j = (1,\dots,1)$. The CDC of a graph $G$ is the direct product $G\times K_2$. The main eigenspace of $\mathbf A$ is generated by the principal main eigenvectors and is the same as the image of the walk matrix. A hierarchy of properties of pairs of graphs is established in view of their CDC's, walk matrices, main eigenvalues, eigenvectors and eigenspaces. We determine by algorithm that there are 32 pairs of non-isomorphic graphs on at most 8 vertices which have the same CDC.

math.CO↗

arXiv2018

Fast Algorithms for Indices of Nested Split Graphs Approximating Real Complex Networks

We present a method based on simulated annealing to obtain a nested split graph that approximates a real complex graph. This is used to compute a number of graph indices using very efficient algorithms that we develop, leveraging the geometrical properties of nested split graphs. Practical results are given for six graphs from such diverse areas as social networks, communication networks, word associations, and molecular chemistry. We present a critical analysis of the appropriate perturbation schemes that search the whole space of nested split graphs and the distance functions that gauge the dissimilarity between two graphs.

cs.DM↗

arXiv2015

Graphs that have a weighted adjacency matrix with spectrum $\{λ_1^{n-2}, λ_1^2\}$

In this paper we completely characterize the graphs which have an edge weighted adjacency matrix belonging to the class of $n \times n$ involutions with spectrum equal to $\{ λ_1^{n-2}, λ_2^{2} \}$ for some $λ_1$ and some $λ_2$. The connected graphs turn out to be the cographs constructed as the join of at least two unions of pairs of complete graphs, and possibly joined with one other complete graph.

math.CO↗