Infinite Family Of Graphs Research Articles

Graphs with large multiplicity of −2 in the spectrum of the eccentricity matrix

The eccentricity matrix of a simple connected graph is obtained from the distance matrix by keeping only the entries that are largest in at least one of their row or column. This matrix can be seen as a counterpart to the standard adjacency matrix, since the latter one is also obtained from the distance matrix but this time by keeping only the entries equal to 1. It is known that, for λ∉{−1,0} and a fixed i∈N, when n passes N there is only a finite number of graphs with n vertices having λ as an eigenvalue of multiplicity n−i in the spectrum of the adjacency matrix. This phenomenon motivates us to consider graphs with large multiplicity of an eigenvalue of the eccentricity matrix. In this context, we determine all connected graphs with n vertices for which −2 has the multiplicity n−i, where i≤5. Infinite families of graphs with this spectral property are encountered. Results of this paper can be compared to results concerning graphs with large multiplicity of zero in the spectrum of the adjacency matrix; at present the graphs for which this multiplicity is n−i, where i≤4, are known. Our results also become meaningful in the framework of the median eigenvalue problem since, for sufficiently large n, the median eigenvalue of the obtained graphs is always −2.

Domination of triangulated discs and maximal outerplanar graphs

Given an infinite family of graphs, its domination ratio is the smallest real k such that for every large enough graph in the family, the ratio between its domination number and order is at most k. We give bounds for the domination ratios of various families of triangulated discs, disproving two conjectures of Tokunaga. For instance, we show that the domination ratio of triangulated discs of minimum degree 3 is at least 3/10. We also give a natural extension of a theorem on the domination number of maximal outerplane graphs, proven by Campos and Wakabayashi, and independently Tokunaga. Motivated by the Matheson-Tarjan Conjecture, we present observations on dominating triangulations via Jackson-Yu decomposition trees as well as an approach on how to efficiently dominate, given a triangulation, many large subtriangulations thereof. Throughout the paper, results are accompanied by computational experiments.

Thrackles, Superthrackles and the Hanani-Tutte Theorem

A thrackle is a drawing of a graph in which any two vertex-disjoint edges cross exactly once and incident edges do not cross. A graph that has a thrackle drawing is thracklable. In the past three decades, mathematicians investigated a number of variations of thrackles by relaxing or changing the required number of crossings on edges or restricting the placement of vertices on the surface (e.g., restricting to outerdrawings, in which vertices are restricted to a boundary curve of the surface). A graph is superthracklable if it can be drawn so that any two edges cross exactly once. We provide a simple proof for the following: a graph can be drawn on the plane so that every two edges cross an odd number of times if and only if it is superthracklable on the plane. A graph is outersuperthracklable if it has a drawing on a disc in which every vertex is on the boundary of the disc and every pair of edges cross exactly once. We introduce some natural generalisations of outersuperthracklable graphs and we show that these classes of graphs are equivalent. Lastly, using the Hanani-Tutte characterisation of planar graphs we show that for any surface $\Sigma$, there is a relationship between the class of graphs that are not embeddable on $\Sigma$ and the class of graphs that are not superthracklable with respect to $\Sigma$. More specifically, we show how to construct, from any forbidden minor $G$ for embeddability in $\Sigma$, two infinite families of graphs that are not superthracklable with respect to $\Sigma$. This sheds further light on the relationship between some of Archdeacon and Stor's forbidden configurations for superthrackles and Kuratowski's forbidden minors for planarity.

Cospectral mates for generalized Johnson and Grassmann graphs

We provide three infinite families of graphs in the Johnson and Grassmann schemes that are not uniquely determined by their spectrum. We do so by constructing graphs that are cospectral but non-isomorphic to these graphs.

Factorizing the Rado graph and infinite complete graphs

Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of infinite graphs, together with $\Lambda$. The Factorization Problem $FP(\mathcal{F}, \Lambda)$ asks whether $\mathcal{F}$ can be realized as a factorization of $\Lambda$, namely, whether there is a factorization $\mathcal{G}=\{\Gamma_{\alpha}: \alpha\in \mathcal{A}\}$ of $\Lambda$ such that each $\Gamma_{\alpha}$ is a copy of $F_{\alpha}$.We study this problem when $\Lambda$ is either the Rado graph $R$ or the complete graph $K_\aleph$ of infinite order $\aleph$. When $\mathcal{F}$ is a countable family, we show that $FP(\mathcal{F}, R)$ is solvable if and only if each graph in $\mathcal{F}$ has no finite dominating set. We also prove that $FP(\mathcal{F}, K_\aleph)$ admits a solution whenever the cardinality of $\mathcal{F}$ coincides with the order and the domination numbers of its graphs.For countable complete graphs, we show some non existence results when the domination numbers of the graphs in $\mathcal{F}$ are finite. More precisely, we show that there is no factorization of $K_\N$ into copies of a $k$-star (that is, the vertex disjoint union of $k$ countable stars) when $k=1,2$, whereas it exists when $k\geq 4$, leaving the problem open for $k=3$.Finally, we determine sufficient conditions for the graphs of a decomposition to be arranged into resolution classes.

A unified criterion for distinguishing graphs by their spectral radius

Complementarity spectrum of a connected graph G, denoted by Π ( G ) , is the set of spectral radii of all connected induced subgraphs of G. Further, G is said to be spectrally non-redundant if c ( G ) , the cardinality of Π ( G ) , is equal to b ( G ) , the number of all non-isomorphic induced subgraphs of G. In this paper, we give a sufficient condition for a family of graphs to be spectrally non-redundant. Using this criterion, we show that several infinite families of graphs are spectrally non-redundant. Moreover, we apply the same condition to distinguish graphs by their spectral radius, which illustrates the main reason for associating a graph with its complementarity spectrum.

Fixing Numbers of Point-Block Incidence Graphs

A vertex in a graph is referred to as fixed if it is mapped to itself under every automorphism of the vertices. The fixing number of a graph is the minimum number of vertices, when fixed, that fixes all of the vertices in the graph. Fixing numbers were first introduced by Laison and Gibbons, and independently by Erwin and Harary. Fixing numbers have also been referred to as determining numbers by Boutin. The main motivation is to remove all symmetries from a graph. A very simple application is in the creation of QR codes where the symbols must be fixed against any rotation. We determine the fixing number for several families of graphs, including those arising from combinatorial block designs. We also present several infinite families of graphs with an even stronger condition, where fixing any vertex in a graph fixes every vertex.

More reliable graphs are not always stronger

AbstractA graph is stronger than a graph if has at least as many connected spanning subgraphs of size as for any positive integer . Counting the number of connected spanning subgraphs of fixed size allows us to compute the reliability of a graph. Formally, the reliability polynomial of a graph is the probability that the graph is connected when each edge is deleted independently with the same fixed probability. A graph is uniformly more reliable than if its reliability polynomial is greater than or equal to the reliability polynomial of for all probabilities. As a direct consequence of the definition, a sufficient condition for to be uniformly more reliable than is for to be stronger than . In this paper, we show that the sufficient condition is not necessary by providing an example of two infinite families of graphs, and , such that is uniformly more reliable than but is not stronger than .

Planar Legendrian Θ ${\rm{\Theta }}$‐graphs

AbstractWe classify topologically trivial Legendrian ‐graphs in and identify the complete family of nondestabilizable Legendrian realizations in this topological class. In contrast to all known results for Legendrian knots, this is an infinite family of graphs. We also show that any planar graph that contains a subdivision of a ‐graph or as a subgraph will have an infinite number of distinct, topologically trivial nondestabilizable Legendrian embeddings. Additionally, we introduce two moves, vertex stabilization and vertex twist, that change the Legendrian type within a smooth isotopy class.

Radius, leaf number, connected domination number and minimum degree

Let G be a simple, connected graph with minimum degree δ, radius r and leaf number L(G). We prove that We give similar bounds for triangle-free graphs. Infinite families of graphs are constructed to show that all the bounds here are sharp, except the one for r ≡ 1 modulo 3 in the above piecewise inequality. The results bring to literature new lower bounds on the leaf number and new upper bounds on the radius and connected domination number of a graph. Further, the techniques applied in this paper can be used to improve known asymptotically sharp bounds on the radius and diameter to sharp bounds. We consider simple graphs only.

Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

The kth power of a graph G=(V,E), Gk, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This article proves various eigenvalue bounds for the independence number and chromatic number of Gk which purely depend on the spectrum of G, together with a method to optimize them. Our bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.

Graphical Frobenius representations of non-abelian groups

A group G has a Frobenius graphical representation (GFR) if there is a simple graph Γ whose full automorphism group is isomorphic to G acting on the vertices as a Frobenius group. In particular, any group G with a GFR is a Frobenius group and Γ is a Cayley graph. By very recent results of Spiga, there exists a function f such that if G is a finite Frobenius group with complement H and | G | > f (| H |) then G admits a GFR. This paper provides an infinite family of graphs that admit GFRs despite not meeting Spiga’s bound. In our construction, the group G is the Higman group A ( f , q 0 ) for an infinite sequence of f and q 0 , having a nonabelian kernel and a complement of odd order.

Laplacian Fractional Revival on Graphs

We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are double cones. Finally, we construct, through Cartesian products and joins, several infinite families of graphs that admit Laplacian fractional revival; some of these graphs exhibit polygamous fractional revival.

Graphs with few distinct eigenvalues and extremal energy

In his survey “Beyond graph energy: Norms of graphs and matrices” [23], Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph, respectively. We show that these graphs have at most two nonzero distinct singular values and investigate the proposed problems organizing our study according to the type of spectrum they can have. In most cases all graphs are characterized. Infinite families of graphs are given otherwise. We also show that all graphs satisfying the properties required in the problems are integral, except for complete bipartite graphs Kp,q and disconnected graphs with a component Kp,q, where pq is not a perfect square.

A note on the metric and edge metric dimensions of 2-connected graphs

For a given graph G , the metric and edge metric dimensions of G , dim ( G ) and edim ( G ) , are the cardinalities of the smallest possible subsets of vertices in V ( G ) such that they uniquely identify the vertices and the edges of G , respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph G such that dim ( G ) = a and edim ( G ) = b for every pair of integers a , b , where 4 ≤ b < a . For this we use subdivisions of complete graphs, whose metric dimension is in some cases smaller than the edge metric dimension. Along the way, we present an upper bound for the metric and edge metric dimensions of subdivision graphs under some special conditions.

Higher discrete homotopy groups of graphs

This paper studies a discrete homotopy theory for graphs introduced by Barcelo et al. We prove two main results. First we show that if G is a graph containing no 3- or 4-cycles, then the nth discrete homotopy group A n (G) is trivial for all n≥2. Second we exhibit for each n≥1 a natural homomorphism ψ:A n (G)→ℋ n (G), where ℋ n (G) is the nth discrete cubical singular homology group, and an infinite family of graphs G for which ℋ n (G) is nontrivial and ψ is surjective. It follows that for each n≥1 there are graphs G for which A n (G) is nontrivial.

Two families of graphs that are Cayley on nonisomorphic groups

A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the groups have order $pq$; when the Cayley graphs are normal; or when the groups are both abelian. In this paper, we construct two infinite families of graphs, each of which is Cayley on an abelian group and a nonabelian group. These families include the smallest examples of such graphs that had not appeared in other results.

Families of Integral Cographs within a Triangular Array

AbstractThe determinant Hosoya triangle, is a triangular array where the entries are the determinants of two-by-two Fibonacci matrices. The determinant Hosoya triangle mod 2 gives rise to three infinite families of graphs, that are formed by complete product (join) of (the union of) two complete graphs with an empty graph. We give a necessary and sufficient condition for a graph from these families to be integral.Some features of these graphs are: they are integral cographs, all graphs have at most five distinct eigenvalues, all graphs are either d-regular graphs with d =2, 4, 6, . . . or almost-regular graphs, and some of them are Laplacian integral. Finally we extend some of these results to the Hosoya triangle.

The average size of a connected vertex set of a graph—Explicit formulas and open problems

AbstractAlthough connectivity is a basic concept in graph theory, the enumeration of connected subgraphs of a graph has only recently received attention. The topic of this paper is the average order of a connected induced subgraph of a graph. This generalizes, to graphs in general, the average order of a subtree of a tree. For various infinite families of graphs, we investigate the asymptotic behavior of the proportion of vertices in an induced connected subgraph of average order. For ladders and circular ladders, an explicit closed formula is derived for the average order of a connected induced subgraph in terms of the classic Pell numbers. These formulas imply that, asymptotically, 3/4 of the vertices of a ladder or circular ladder, on average, are present in a connected induced subgraph. Results on such infinite families motivate an assortment of open problems.

A new class of polynomials from the spectrum of a graph, and its application to bound the k-independence number

The k-independence number of a graph is the maximum size of a set of vertices at pairwise distance greater than k. A graph is called k-partially walk-regular if the number of closed walks of a given length l≤k, rooted at a vertex v, only depends on l. In particular, a distance-regular graph is also k-partially walk-regular for any k. In this paper, we introduce a new family of polynomials obtained from the spectrum of a graph, called minor polynomials. These polynomials, together with the interlacing technique, allow us to give tight spectral bounds on the k-independence number of a k-partially walk-regular graph. With some examples and infinite families of graphs whose bounds are tight, we also show that the odd graph Oℓ with ℓ odd has no 1-perfect code. Moreover, we show that our bound coincides, in general, with the Delsarte's linear programming bound and the Lovász theta number θ, the best ones to our knowledge. In fact, as a byproduct, it is shown that the given bounds also apply for θ and Θ, the Shannon capacity of a graph. Moreover, apart from the possible interest that the minor polynomials can have, our approach has the advantage of yielding closed formulas and, so, allowing asymptotic analysis.