On the distance spectrum and distance-based topological indices of central vertex-edge join of three graphs

Topological indices are molecular descriptors that describe the properties of chemical compounds. These topological indices correlate specific physico-chemical properties like boiling point, enthalpy of vaporization, strain energy, and stability of chemical compounds. This article introduces a new graph operation based on central graph called central vertex-edge join and provides its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, and Wiener index. Also, we discuss the distance spectrum of the central vertex-edge join of three regular graphs. Furthermore, we obtain new families of $D$-equienergetic graphs, which are non $D$-cospectral

Distance matrices on the H-join of graphs: A general result and applications

Given a graph $H$ with vertices $1,\ldots ,s$ and a set of pairwise vertex disjoint graphs $G_{1},\ldots ,G_{s},$ the vertex $i$ of $H$ is assigned to $G_{i}.$ Let $G$ be the graph obtained from the graphs $G_{1},\ldots ,G_{s}$ and the edges connecting each vertex of $G_{i}$ with all the vertices of $G_{j}$ for all edge $ij$ of $H.$ The graph $G$ is called the $H-join$ of $G_1,\ldots,G_s$. Let $M(G)$ be a matrix on a graph $G$. A general result on the eigenvalues of $M\left( G\right)$, when the all ones vector is an eigenvector of $M\left( G_{i}\right)$ for $i=1,2,\ldots ,s$, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of $G$ when $G_{1},\ldots ,G_{s}$ are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.publishe

Small-worlds: How and why

We investigate small-world networks from the point of view of their origin. While the characteristics of small-world networks are now fairly well understood, there is as yet no work on what drives the emergence of such a network architecture. In situations such as neural or transportation networks, where a physical distance between the nodes of the network exists, we study whether the small-world topology arises as a consequence of a tradeoff between maximal connectivity and minimal wiring. Using simulated annealing, we study the properties of a randomly rewired network as the relative tradeoff between wiring and connectivity is varied. When the network seeks to minimize wiring, a regular graph results. At the other extreme, when connectivity is maximized, a near random network is obtained. In the intermediate regime, a small-world network is formed. However, unlike the model of Watts and Strogatz (Nature {\bf 393}, 440 (1998)), we find an alternate route to small-world behaviour through the formation of hubs, small clusters where one vertex is connected to a large number of neighbours.Comment: 20 pages, latex, 9 figure

Local resilience and Hamiltonicity Maker-Breaker games in random-regular graphs

For an increasing monotone graph property \mP the \emph{local resilience} of a graph $G$ with respect to \mP is the minimal $r$ for which there exists of a subgraph $H\subseteq G$ with all degrees at most $r$ such that the removal of the edges of $H$ from $G$ creates a graph that does not possesses \mP. This notion, which was implicitly studied for some ad-hoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the Binomial random graph model \GNP and some families of pseudo-random graphs with respect to several graph properties such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudo-random \emph{regular} graphs of constant degree. We investigate the local resilience of the typical random $d$-regular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular we prove that for every positive $\epsilon$ and large enough values of $d$ with high probability the local resilience of the random $d$-regular graph, \GND, with respect to being Hamiltonian is at least $(1-\epsilon)d/6$. We also prove that for the Binomial random graph model \GNP, for every positive $\epsilon>0$ and large enough values of $K$, if $p>\frac{K\ln n}{n}$ then with high probability the local resilience of \GNP with respect to being Hamiltonian is at least $(1-\epsilon)np/6$. Finally, we apply similar techniques to Positional Games and prove that if $d$ is large enough then with high probability a typical random $d$-regular graph $G$ is such that in the unbiased Maker-Breaker game played on the edges of $G$, Maker has a winning strategy to create a Hamilton cycle.Comment: 34 pages. 1 figur

The isoperimetric problem in Johnson graphs

It has been recently proved that the connectivity of distance regular graphs is the degree of the graph. We study the Johnson graphs $J(n,m)$, which are not only distance regular but distance transitive, with the aim to analyze deeper connectivity properties in this class. The vertex $k$-connectivity of a graph $G$ is the minimum number of vertices that have to be removed in order to separate the graph into two sets of at least $k$ vertices in each one. The isoperimetric function $\mu_G(k)$ of a graph $G$ is the minimum boundary among all subsets of vertices of fixed cardinality $k$. We give the value of the isoperimetric function of the Johnson graph $J(n,m)$ for values of $k$ of the form ${t\choose m}$, and provide lower and upper bounds for this function for a wide range of its parameter. The computation of the isoperimetric function is used to study the $k$-connectivity of the Johnson graphs as well. We will see that the $k$-connectivity grows very fast with $k$, providing much sensible information about the robustness of these graphs than just the ordinary connectivity. In order to study the isoperimetric function of Johnson graphs we use combinatorial and spectral tools. The combinatorial tools are based on compression techniques, which allow us to transform sets of vertices without increasing their boundary. In the compression process we will show that sets of vertices that induce Johnson subgraphs are optimal with respect to the isoperimetric problem. Upper bounds are obtained by displaying nested families of sets which interpolate optimal ones. The spectral tools are used to obtain lower bounds for the isoperimetric function. These tools allow us to display completely the isoperimetric function for Johnson graphs $J(n,3)$. . Distance regular graphs form a structured class of graphs which include well-known families, as the n-cubes. Isoperimetric inequalities are well understood for the cubes, but for the general class of distance regular class it has only been proved that the connectivity of these graphs equals the degree. As another test case, the project suggests to study the family of so-called Johnson graphs, which are not only distance regular but also distance transitive. Combinatorial and spectral techniques to analyze the problem are available

Long cycles and paths in distance graphs

AbstractFor n∈N and D⊆N, the distance graph PnD has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, PnD has a component of order at least n−cD if and only if for every n≥cD+3, PnD has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result