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Graph theory is a useful mathematical structure used to model pairwise relations between sensor nodes in wireless sensor networks. Graph equations are nothing but equations in which the unknown factors are graphs. Many problems and results in graph theory can be formulated in terms of graph equations. In this paper, we solved some graph equations of detour two-distance graphs, detour three-distance graphs, detour antipodal graphs involving with the line graphs.

This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various “nonpositive curvature\" and “local-to-global” properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises ( We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With

We consider a random planar map M n which is uniformly distributed over the class of all rooted q-angulations with n faces. We let m n be the vertex set of M n , which is equipped with the graph distance d gr . Both when q ≥ 4 is an even integer and when q = 3, there exists a positive constant c q such that the rescaled metric spaces (m n , c q n -1/4 d gr ) converge in distribution in the Gromov—Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.

In this article, we compute the distance, distance Laplacian, and distance signless Laplacian eigenvalues of central vertex and central edge join of two graphs G 1 and G 2 , when G 1 is triangle-free regular and G 2 is regular. These results enable us to determine infinitely many pairs of distance, distance Laplacian and distance signless Laplacian cospectral graphs. In addition, we obtain some lower and upper bounds for the distance spectral radius of the central graph of a triangle-free regular graph. As an application, we construct some new classes of non D -cospectral D -equienergetic graphs.

A connected graph G of diameter diam ( G ) ≥ ℓ is ℓ -distance-balanced if | W xy | = | W yx | for every x , y ∈ V ( G ) with d G ( x , y ) = ℓ , where W xy is the set of vertices of G that are closer to x than to y . We prove that the generalized Petersen graph GP ( n ,  k ) is diam ( G P ( n , k ) ) -distance-balanced provided that n is large enough relative to k . This partially solves a conjecture posed by Miklavič and Šparl (Discrete Appl Math 244:143–154, 2018). We also determine diam ( G P ( n , k ) ) when n is large enough relative to k .

For an edge $uv$ in a graph $G$ , $W_{u,v}^{G}$ denotes the set of all vertices of $G$ that are closer to $u$ than to $v$ . A graph $G$ is said to be quasi-distance-balanced if there exists a constant $\\unicode[STIX]{x1D706}>1$ such that $|W_{u,v}^{G}|=\\unicode[STIX]{x1D706}^{\\pm 1}|W_{v,u}^{G}|$ for every pair of adjacent vertices $u$ and $v$ . The existence of nonbipartite quasi-distance-balanced graphs is an open problem. In this paper we investigate the possible structure of cycles in quasi-distance-balanced graphs and generalise the previously known result that every quasi-distance-balanced graph is triangle-free. We also prove that a connected quasi-distance-balanced graph admitting a bridge is isomorphic to a star. Several open problems are posed.

Quantifying the differences between networks is a challenging and ever-present problem in network science. In recent years, a multitude of diverse, ad hoc solutions to this problem have been introduced. Here, we propose that simple and well-understood ensembles of random networks—such as Erdős–Rényi graphs, random geometric graphs, Watts–Strogatz graphs, the configuration model and preferential attachment networks—are natural benchmarks for network comparison methods. Moreover, we show that the expected distance between two networks independently sampled from a generative model is a useful property that encapsulates many key features of that model. To illustrate our results, we calculate this within-ensemble graph distance and related quantities for classic network models (and several parameterizations thereof) using 20 distance measures commonly used to compare graphs. The within-ensemble graph distance provides a new framework for developers of graph distances to better understand their creations and for practitioners to better choose an appropriate tool for their particular task.

A matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on n vertices is ⌊3n-12n-3⌋. In this paper we prove this conjecture for all n≥1. The main geometric ingredient of the proof is an isoperimetric inequality related to L’Huilier’s inequality.

Consider a set of n vertices, where each vertex has a location in $\\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\\mathbb{R}^d$ , and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations and the vertex weights. Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on $\\mathbb{R}^d$ with vertex locations given by a homogeneous Poisson point process, having weights which are independent and identically distributed copies of limiting vertex weights. Our set-up covers many sparse geometric random graph models from the literature, including geometric inhomogeneous random graphs (GIRGs), hyperbolic random graphs, continuum scale-free percolation, and weight-dependent random connection models. We prove that the limiting degree distribution is mixed Poisson and the typical degree sequence is uniformly integrable, and we obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a byproduct of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

An odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle. We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane so that the lengths of the edges are odd integers.