Families of graphs with twin pendent paths and the Braess edge

In the context of a random walk on an undirected graph, Kemeny's constant can measure the average travel time for a random walk between two randomly chosen vertices. We are interested in graphs that behave counter-intuitively in regard to Kemeny's constant: in particular, we examine graphs with a cut-vertex at which at least two branches are paths, regarding whether the insertion of a particular edge into a graph results in an increase of Kemeny's constant. We provide several tools for identifying such an edge in a family of graphs and for analysing asymptotic behaviour of the family regarding the tendency to have that edge; and classes of particular graphs are given as examples. Furthermore, asymptotic behaviours of families of trees are described.

Some Bond Incident Degree Indices of (Molecular) Graphs with Fixed Order and Number of Cut Vertices

A bond incident degree (BID) index of a graph G is defined as ∑ u v ∈ E G f d G u , d G v , where d G w denotes the degree of a vertex w of G , E G is the edge set of G , and f is a real-valued symmetric function. The choice f d G u , d G v = a d G u + a d G v in the aforementioned formula gives the variable sum exdeg index SEI a , where a ≠ 1 is any positive real number. A cut vertex of a graph G is a vertex whose removal results in a graph with more components than G has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by V n , k the class of all n -vertex graphs with k ≥ 1 cut vertices and containing at least one cycle. Recently, Du and Sun [AIMS Mathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of SEI a from the set V n k for a > 1 . In the present paper, we not only characterize the graphs with the minimum value of SEI a from the set V n k for a > 1 , but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all n -vertex molecular graphs with k ≥ 1 cut vertices and containing at least one cycle.

Cyclic connectivity index of fuzzy incidence graphs with applications in the highway system of different cities to minimize road accidents and in a network of different computers

A parameter is a numerical factor whose values help us to identify a system. Connectivity parameters are essential in the analysis of connectivity of various kinds of networks. In graphs, the strength of a cycle is always one. But, in a fuzzy incidence graph (FIG), the strengths of cycles may vary even for a given pair of vertices. Cyclic reachability is an attribute that decides the overall connectedness of any network. In graph the cycle connectivity (CC) from vertex a to vertex b and from vertex b to vertex a is always one. In fuzzy graph (FG) the CC from vertex a to vertex b and from vertex b to vertex a is always same. But if someone is interested in finding CC from vertex a to an edge ab, then graphs and FGs cannot answer this question. Therefore, in this research article, we proposed the idea of CC for FIG. Because in FIG, we can find CC from vertex a to vertex b and also from vertex a to an edge ab. Also, we proposed the idea of CC of fuzzy incidence cycles (FICs) and complete fuzzy incidence graphs (CFIGs). The fuzzy incidence cyclic cut-vertex, fuzzy incidence cyclic bridge, and fuzzy incidence cyclic cut pair are established. A condition for CFIG to have fuzzy incidence cyclic cut-vertex is examined. Cyclic connectivity index and average cyclic connectivity index of FIG are also investigated. Three different types of vertices, such as cyclic connectivity increasing vertex, cyclically neutral vertex and, cyclic connectivity decreasing vertex, are also defined. The real-life applications of CC of FIG in a highway system of different cities to minimize road accidents and a computer network to find the best computers among all other computers are also provided.

Menger’s theorem for m-polar fuzzy graphs and application of m-polar fuzzy edges to road network

The main objective of this research article is to classify different types of m-polar fuzzy edges in an m-polar fuzzy graph by using the strength of connectedness between pairs of vertices. The identification of types of m-polar fuzzy edges, including α-strong m-polar fuzzy edges, β-strong m-polar fuzzy edges and δ-weak m-polar fuzzy edges proved to be very useful to completely determine the basic structure of m-polar fuzzy graph. We analyze types of m-polar fuzzy edges in strongest m-polar fuzzy path and m-polar fuzzy cycle. Further, we define various terms, including m-polar fuzzy cut-vertex, m-polar fuzzy bridge, strength reducing set of vertices and strength reducing set of edges. We highlight the difference between edge disjoint m-polar fuzzy path and internally disjoint m-polar fuzzy path from one vertex to another vertex in an m-polar fuzzy graph. We define strong size of an m-polar fuzzy graph. We then present the most celebrated result due to Karl Menger for m-polar fuzzy graphs and illustrate the vertex version of Menger’s theorem to find out the strongest m-polar fuzzy paths between affected and non-affected cities of a country due to an earthquake. Moreover, we discuss an application of types of m-polar fuzzy edges to determine traffic-accidental zones in a road network. Finally, a comparative analysis of our research work with existing techniques is presented to prove its applicability and effectiveness.

Nonessential sum graph of an Artinian ring

PurposeThe authors study the interdisciplinary relation between graph and algebraic structure ring defining a new graph, namely “non-essential sum graph”. The nonessential sum graph, denoted by NES(R), of a commutative ring R with unity is an undirected graph whose vertex set is the collection of all nonessential ideals of R and any two vertices are adjacent if and only if their sum is also a nonessential ideal of R.Design/methodology/approachThe method is theoretical.FindingsThe authors obtain some properties of NES(R) related with connectedness, diameter, girth, completeness, cut vertex, r-partition and regular character. The clique number, independence number and domination number of NES(R) are also found.Originality/valueThe paper is original.

On Maximal Distance Energy

Let G be a graph of order n. If the maximal connected subgraph of G has no cut vertex then it is called a block. If each block of graph G is a clique then G is called clique tree. The distance energy ED(G) of graph G is the sum of the absolute values of the eigenvalues of the distance matrix D(G). In this paper, we study the properties on the eigencomponents corresponding to the distance spectral radius of some special class of clique trees. Using this result we characterize a graph which gives the maximum distance spectral radius among all clique trees of order n with k cliques. From this result, we confirm a conjecture on the maximum distance energy, which was given in Lin et al. Linear Algebra Appl 467(2015) 29-39.

On the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph

Let \(G=(V,E)\) be a graph. Then the first and second entire Zagreb indices of \(G\) are defined, respectively, as \(M_{1}^{\varepsilon}(G)=\displaystyle \sum_{x \in V(G) \cup E(G)} (d_{G}(x))^{2}\) and \(M_{2}^{\varepsilon}(G)=\displaystyle \sum_{\{x,y\}\in B(G)} d_{G}(x)d_{G}(y)\), where \(B(G)\) denotes the set of all 2-element subsets \(\{x,y\}\) such that \(\{x,y\} \subseteq V(G) \cup E(G)\) and members of \(\{x,y\}\) are adjacent or incident to each other. In this paper, we obtain the entire Zagreb indices of the line graph and line cut-vertex graph of the subdivision graph of the friendship graph.

Connected Soft Graph

The theory of soft set offers a mathematical tool to deal with uncertainty. Nowadays, work on soft graph theory is progressing rapidly. In this paper, we define connected soft graph and derive some results. We also define cut vertex and bridge of soft graph along with some results on it.