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A chain graph is a bipartite graph in which the neighborhood of the vertices in each partite set forms a chain with respect to set inclusion. By extending the concept of nesting from a bipartite graph to a k partite graph, a k-nested graph is defined. A half graph is a chain graph having no pairs of duplicate vertices. Similarly, a 'k-half graph' is a class of knested graph with no pairs of duplicate vertices. The (first) Zagreb matrix or Z-matrix denoted by Z(G) = (zij)n×n of a graph G, whose vertex vi has degree di is defined by zij = di + dj if the vertices vi and vj are adjacent and zij = 0 otherwise. Let ζ1, ζ2, . . ., ζn be the Zagreb eigenvalues of Z(G) and the Zagreb energy is the sum of the absolute values of the Zagreb eigenvalues. We obtain the determinant, eigenvalues and inverse of a k-half graph with respect to the Z-matrix. Bounds for the Zagreb energy and spectral radius are discussed along with the main and non-main Zagreb eigenvalues of a k-half graph. [ABSTRACT FROM AUTHOR]

For a graph, the first (multiplicative) Zagreb index is equal to the sum (product) of squares of the vertex degrees, and the second (multiplicative) Zagreb index is equal to the sum (product) of products of the degrees of a pair of adjacent vertices. In this work, by a unified approach, we determine the extremal values of these Zagreb indices in terms of the (edge) connectivity and characterize the corresponding extremal graphs among all connected bipartite graphs of order n. Our results show that the extremal graphs of given (edge) connectivity regarding the Zagreb indices and multiplicative Zagreb indices do not completely coincide with other topological indices. [ABSTRACT FROM AUTHOR]

A two-mode network is a type of network in which nodes can be divided into two sets in such a way that links can be established between different types of nodes. The relationship between two separate sets of entities can be modeled as a bipartite network. In computer networks data is transmitted in form of packets between source to destination. Such packet-switched networks rely on routing protocols to select the best path. Configurations of these protocols depends on the network acquirements; that is why one routing protocol might be efficient for one network and may be inefficient for a other. Because some protocols deal with hop-count (number of nodes in the path) while others deal with distance vector. This paper investigates the minimum transmission in two-mode networks. Based on some parameters, we obtained the minimum transmission between the class of all connected n-nodes in bipartite networks. These parameters are helpful to modify or change the path of a given network. Furthermore, by using least squares fit, we discussed some numerical results of the regression model of the boiling point in benzenoid hydrocarbons. The results show that the correlation of the boiling point in benzenoid hydrocarbons of the first Zagreb eccentricity index gives better result as compare to the correlation of second Zagreb eccentricity index. In case of a connected network, the first Zagreb eccentricity index ξ 1 (ℵ) is defined as the sum of the square of eccentricities of the nodes, and the second Zagreb eccentricity index ξ 2 (ℵ) is defined as the sum of the product of eccentricities of the adjacent nodes. This article deals with the minimum transmission with respect to ξ i (ℵ) , for i = 1 , 2 among all n-node extremal bipartite networks with given matching number, diameter, node connectivity and link connectivity. [ABSTRACT FROM AUTHOR]

Let G be a simple graph. The subdivision graph and the double graph are the graphs obtained from a given graph G which have several properties related to the properties of G. In this paper, the first and second Zagreb and multiplicative Zagreb indices of double graphs, subdivision graphs, double graphs of the subdivision graphs and subdivision graphs of the double graphs of G are obtained. In particular, these numbers are calculated for the frequently used null, path, cycle, star, complete, complete bipartite or tadpole graph. [ABSTRACT FROM AUTHOR]

Classical topological indices, such as Zagreb indices ( M 1 and M 2 ) and the well-studied eccentric connectivity index ( ξ c ) directly or indirectly consider the total contribution of all edges in a graph. By considering the total degree sum of all non-adjacent vertex pairs in a graph, Ashrafi et al. (Discrete Appl Math 158:1571–1578, 2010) proposed two new Zagreb-type indices, namely the first Zagreb coindex ( M ¯ 1 ) and second Zagreb coindex ( M ¯ 2 ), respectively. Motivated by Ashrafi et al., we consider the total eccentricity sum of all non-adjacent vertex pairs, which we call the eccentric connectivity coindex ( ξ ¯ c ), of a connected graph. In this paper, we study the extremal problems of ξ ¯ c for connected graphs of given order, connected graphs of given order and size, and the trees, unicyclic graphs, bipartite graphs containing cycles and triangle-free graphs of given order, respectively. Additionally, we establish various lower bounds for ξ ¯ c in terms of several other graph parameters. [ABSTRACT FROM AUTHOR]

Let G = (V , E) be a connected graph, and d (u) the degree of vertex u ∈ V. We define the general Z -type index of G as Z α , β (G) = ∑ u v ∈ E [ d (u) + d (v) − β ] α , where α and β are two real numbers. This generalizes several famous topological indices, such as the first and second Zagreb indices, the general sum-connectivity index, the reformulated first Zagreb index, and the general Platt index, which have successful applications in QSPR/QSAR research. Hence, we are able to study these indices in a unified approach. Let C (π) the set of connected graphs with degree sequence π. In the present paper, under different conditions of α and β , we show that: (1) There exists a so-called BFS-graph having extremal Z α , β index in C (π) ; (2) If π is the degree sequence of a tree, a unicyclic graph, or a bicyclic graph, with minimum degree 1, then there exists a special BFS-graph with extremal Z α , β index in C (π) ; (3) The so-called majorization theorem of Z α , β holds for trees, unicyclic graphs, and bicyclic graphs. As applications of the above results, we determine the extremal graphs with maximum Z α , β index for α > 1 and β ≤ 2 in the set of trees, unicyclic graphs, and bicyclic graphs with given number of pendent vertices, maximum degree, independence number, matching number, and domination number, respectively. These extend the main results of some published papers. [ABSTRACT FROM AUTHOR]