Monophonic Distance in Graphs

For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance dm(u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is em(v) = max {dm(u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter

Monophonic Distance Laplacian Energy of Transformation Graphs Sn^++-,Sn^{+-+},Sn^{+++}

Let $G$ be a simple connected graph of order $n$, $v_{i}$ its vertex. Let $\delta^{L}_{1}, \delta^{L}_{2}, \ldots, \delta^{L}_{n}$ be the eigenvalues of the distance Laplacian matrix $D^{L}$ of $G$. The distance Laplacian energy is denoted by $LE_{D}(G)$. This motivated us to defined the new graph energy monophonic distance Laplacian energy of graphs. The eigenvalues of monophonic distance Laplacian matrix $M^{L}\left(G\right)$ are denoted by $\mu^{L}_{1}, \mu^{L}_{2}, \ldots, \mu^{L}_{n}$ and are said to be $M^{L}$- eigenvalues of $G$ and to form the $M^{L}$-spectrum of $G$, denoted by $Spec_{M^{L}}(G)$. Here $MT_{G}\left(v_{j}\right)$ is the $j^{th}$ row sum of monophonic distance matrix of $M(G)$ and $\mu^{L}_{1}\leq\mu^{L}_{2}\leq, \ldots, \leq\mu^{L}_{n}$ be the eigenvalues of the monophonic distance Laplacian matrix is $M^{L}(G)$. The monophonic distance Laplacian energy is defined as $LE_{M}(G)$. In this paper we computed the monophonic distance Laplacian energy of $S^{++-}_{n}$, $S^{+-+}_{n}$, $S^{+++}_{n}$ graphs based on its spectrum values.

Byzantine Approximate Agreement on Graphs

Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that 1) the output values are in the convex hull of the non-faulty processors\u27 input values, 2) the output values are within distance d of each other. Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1. In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures

On geodesic and monophonic convexity

In this paper we deal with two types of graph convexities, which are the most natural path convexities in a graph and which are defined by a system P of paths in a connected graph G: the geodesic convexity (also called metric convexity) which arises when we consider shortest paths, and the monophonic convexity (also called minimal path convexity) when we consider chordless paths. First, we present a realization theorem proving, that there is no general relationship between monophonic and geodetic hull sets. Second, we study the contour of a graph, showing that the contour must be monophonic. Finally, we consider the so-called edge Steiner sets. We prove that every edge Steiner set is edge monophonic.Ministerio de Ciencia y TecnologíaFondo Europeo de Desarrollo RegionalGeneralitat de Cataluny

On the monophonic and monophonic domination polynomial of a graph

A set S of vertices of a graph G is a monophonic set of G if each vertex u of G lies on an u − v monophonic path in G for some u, v ∈ S. M ⊆ V (G) is said to be a monophonic dominating set if it is both a monophonic set and a dominating set. Let M(G, i) be the family of monophonic sets of a graph G with cardinality i and let m(G, i) = |M(G, i)|. Then the monophonic polynomial M(G, x) of G is defined as M(G, x) = Ʃⁿi=m(G) m(G, i)xⁱ, where m(G) is the monophonic number of G. In this article, we have introduced monophonic domination polynomial of a graph. We have computed the monophonic and monophonic domination polynomials of some specific graphs. In addition, monophonic and monophonic domination polynomial of the Corona product of two graphs is derived.Publisher's Versio