Detour Polynomials of Generalized Vertex Identified of Graphs

تعد مسافة الالتفاف من أهم أنواع المسافات التي لها تطبيقات حديثة في الكيمياء وشبكات الكمبيوتر، لذلك حصلنا في هذا البحث على متعددات حدود الالتفاف وأدلتها لـ  nمن البيانات المنفصلة عن بعضها البعض بالنسبة للرؤوس ، n≥3. أيضًا وجدنا متعددات حدود الالتفاف وأدلتها لبعض البيانات الخاصة والتي لها تطبيقات مهمة في الكيمياء.The Detour distance is one of the most common distance types used in chemistry and computer networks today. Therefore, in this paper, the detour polynomials and detour indices of vertices identified of n-graphs which are connected to themselves and separated from each other with respect to the vertices for n≥3 will be obtained. Also, polynomials detour and detour indices will be found for another graphs which have important applications in Chemistry.

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

Mathematical Properties of Molecular Descriptors Based on Distances

A survey of a number of molecular descriptors based on distance matrices and distance eigenvalues is given. The following distance matrices are considered: the standard distance matrix, the reverse distance matrix, the complementary distance matrix, the resistance-distance matrix, the detour matrix, the reciprocal distance matrix, the reciprocal reverse Wiener matrix and the reciprocal complementary distance matrix. Mathematical properties are discussed for the following molecular descriptors with a special emphasis on their upper and lower bounds: the reverse Wiener index, the Harary index, the reciprocal reverse Wiener index, the reciprocal complementary Wiener index, the Kirchhoff index, the detour index, the Balaban index, the reciprocal Balaban index, the reverse Balaban index and the largest eigenvalues of distance matrices. This set of molecular descriptors found considerable use in QSPR and QSAR

The Wiener Polynomial Derivatives and Other Topological Indices in Chemical Research

Wiener polynomial derivatives and some other information and topological indices are investigated with respect to their discriminating power and property correlating ability

Detour spectrum and detour energy of conjugate graph complement of dihedral group

Study of graph from a group has become an interesting topic until now. One of the topics is spectra of a graph from finite group. Spectrum of a finite graph is defined as collection of all distinct eigenvalues and their algebraic multiplicity of its matrix. The most related topic in the study of spectrum of finite graph is energy. Energy of a finite graph is defined as sum of absolute value of all its eigenvalues. In this paper, we study the spectrum and energy of detour matrix of conjugate graph complement of dihedral group. The main result is presented as theorems with complete proof

The detour hull number of a graph

For vertices u and v in a connected graph G = (V, E), the set ID[u, v] consists of all those vertices lying on a u−v longest path in G. Given a set S of vertices of G, the union of all sets ID[u, v] for u, v ∈ S, is denoted by ID[S]. A set S is a detour convex set if ID[S] = S. The detour convex hull [S]D of S in G is the smallest detour convex set containing S. The detour hull number dh(G) is the minimum cardinality among the subsets S of V with [S]D = V. A set S of vertices is called a detour set if ID[S] = V. The minimum cardinality of a detour set is the detour number dn(G) of G. A vertex x in G is a detour extreme vertex if it is an initial or terminal vertex of any detour containing x. Certain general properties of these concepts are studied. It is shown that for each pair of positive integers r and s, there is a connected graph G with r detour extreme vertices, each of degree s. Also, it is proved that every two integers a and b with 2 ≤ a ≤ b are realizable as the detour hull number and the detour number respectively, of some graph. For each triple D, k and n of positive integers with 2 ≤ k ≤ n − D + 1 and D ≥ 2, there is a connected graph of order n, detour diameter D and detour hull number k. Bounds for the detour hull number of a graph are obtained. It is proved that dn(G) = dh(G) for a connected graph G with detour diameter at most 4. Also, it is proved that for positive integers a, b and k ≥ 2 with a < b ≤ 2a, there exists a connected graph G with detour radius a, detour diameter b and detour hull number k. Graphs G for which dh(G) = n − 1 or dh(G) = n − 2 are characterized

International Journal of Mathematical Combinatorics, Vol.2A

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group