Friendship Graph Research Articles

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.

On the zero forcing number and propagation time of oriented graphs

Zero forcing is a process of coloring in a graph in time steps known as propagation time. These graph-theoretic parameters have diverse applications in computer science, electrical engineering and mathematics itself. The problem of evaluating these parameters for a network is known to be NP-hard. Therefore, it is interesting to study these parameters for special families of networks. Perila et al. (2017) studied properties of these parameters for some basic oriented graph families such as cycles, stars and caterpillar networks. In this paper, we extend their study to more non-trivial structures such as oriented wheel graphs, fan graphs, friendship graphs, helm graphs and generalized comb graphs. We also investigate the change in propagation time when the orientation of one edge is flipped.

The Maximum Spectral Radius of Graphs Without Friendship Subgraphs

A graph on $2k+1$ vertices consisting of $k$ triangles which intersect in exactly one common vertex is called a $k-$friendship graph and denoted by $F_k$. This paper determines the graphs of order $n$ that have the maximum (adjacency) spectral radius among all graphs containing no $F_k$, for $n$ sufficiently large.

Maximal graphs with respect to rank

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. A reduced graph G is said to be maximal if any reduced graph containing G as a proper induced subgraph has a higher rank. The main intent of this paper is to present some results on maximal graphs. First, we introduce a characterization of maximal trees (a reduced tree is a maximal tree if it is not a proper subtree of a reduced tree with the same rank). Next, we give a near-complete characterization of maximal ‘generalized friendship graphs.’ Finally, we present an enumeration of all maximal graphs with ranks 8 and 9. The ranks up to 7 were already done by Lepović (1990), Ellingham (1993), and Lazić (2010).

Discovering patterns of customer financial behavior using social media data

Social networks are a sterling source of information that reflects the real life of people in the digital space. This makes it possible to infer various aspects of the socioeconomic behavior of the user, even if he/she does not indicate them explicitly. In this study, on the one hand, we consider Russian online social network VK.com, which is analog to the global Facebook platform. On the other hand, there is a supplementary financial information source provided by the bank company. Combining the data of online social media with debit card transactions, we train machine learning models to infer the socioeconomic status (SES) of the user, as well as six purchasing patterns that characterize customer transactional activity of certain type. Namely, we detect if a user is a driver, parent, gamer, traveler, or he/she prefers to purchase at night/in the morning. SES is defined as average monthly expenses and considered as real number variable. The following features are extracted as predictors: demographic information from a user’s page, user participation in communities, topics of that communities, text embeddings of user posts, topological characteristics, and graph embeddings of nodes in the friendship graph. Obtained results show the superiority of graph embeddings in both classification and regression tasks (median absolute percentage error MedAPE = 29.7 for SES). Moreover, for drivers (Macro- $$F_1=0.688$$ ) and parents (Macro- $$F_1=0.679$$ ), the higher scores are reached by concatenation of different features. In addition, we investigate feature importance values and found that topics of user communities and the structure of its network influence on the model stronger than other features. The performed study shows the power of online social media data for inferring user socioeconomic attributes.

Majorization, degree sequence and [formula omitted]-spectral characterization of graphs

Let π=(d1,d2,…,dn) and π′=(d1′,d2′,…,dn′) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π≺π′, if and only if ∑i=1ndi=∑i=1ndi′ and ∑i=1jdi≤∑i=1jdi′ for j=1,2,…,n−1. By using majorization, we show that some graphs are determined by their Aα-spectra. Firstly, we show that the lollipop graph is determined by its Aα-spectrum for 0<α<1, which generalizes the result of Zhang, Liu, Zhang and Yong’s [The lollipop graph is determined by its Q-spectrum, Discrete Math. 309 (2009) 3364–3369]. Secondly, following the result of Cámara and Haemers’ [Spectral characterizations of almost complete graphs, Discrete Appl. Math. 176 (2014) 19–23], we show Kn∖H is determined by its Aα-spectrum where H∈{Pk,K1,k,K2p+q∖pK2} and 1∕2<α<1, which implies that the friendship graph is determined by its Aα-spectrum for 1∕2<α<1.

Double Domination Number of Some Families of Graph

In a graph G = (V, E) each vertex is said to dominate every vertex in its closed neighborhood. In a graph G, if S is a subset of V then S is a double dominating set of G if every vertex in V is dominated by at least two vertices in S. The smallest cardinality of a double dominating set is called the double domination number γx2 (G). [4]. In this paper, we computed some relations between double domination number, domination number, number of vertices (n) and maximum degree (Δ) of Helm graph, Friendship graph, Ladder graph, Circular Ladder graph, Barbell graph, Gear graph and Firecracker graph.

On The Locating-Chromatic Numbers of Subdivisions of Friendship Graph

Let c be a k-coloring of a connected graph G and let pi={C1,C2,...,Ck} be the partition of V(G) induced by c. For every vertex v of G, let c_pi(v) be the coordinate of v relative to pi, that is c_pi(v)=(d(v,C1 ),d(v,C2 ),...,d(v,Ck )), where d(v,Ci )=min{d(v,x)|x in Ci }. If every two vertices of G have different coordinates relative to pi, then c is said to be a locating k-coloring of G. The locating-chromatic number of G, denoted by chi_L (G), is the least k such that there exists a locating k-coloring of G. In this paper, we determine the locating-chromatic numbers of some subdivisions of the friendship graph Fr_t, that is the graph obtained by joining t copies of 3-cycle with a common vertex, and we give lower bounds to the locating-chromatic numbers of few other subdivisions of Fr_t.

MULTIPLICATIVE STATUS NEIGHBORHOOD INDICES OF GRAPHS

Many distance based indices of a graph have been studied in the literature. In this study, we introduce the multiplicative first and second status neighborhood indices, multiplicative sum and product connectivity status neighborhood indices, general multiplicative first and second status neighborhood indices, multiplicative Fstatus neighborhood index, general multiplicative status neighborhood index of a graph and compute exact formulas for some standard graphs and friendship graphs.

Irregular colorings of friendship graph families

In this paper, we acquired the irregular chromatic number for the following graphs: M( fn ), T( fn ), C( fn ), L( fn ), M(DFn ), T(DFn ), L(DFn ), C(DFn ), M(SFn ), T(SFn ), L(SFn ), C(SFn ).

Divisor cordial labeling in the Context of Friendship Graph

Divisor cordial labeling in the Context of Friendship Graph

Characteristic polynomial of anti-adjacency matrix of directed cyclic friendship graph

Graph theory has some applications. One of them is used to do social network analysis as in Facebook with each person as nodes and every like, share, comment, tag as edges. Usually a network can be represented by a graph. Analysing a network is the same as analysing the structure of a graph. A structure of a graph can be analysed through some matrix representations such as anti-adjacency matrix. The entries of the anti-adjacency matrix of a directed graph represent the presence or absence of directed arcs from a vertex to the others. This paper is about the properties of the anti-adjacency matrix of directed cyclic friendship graph, those are the characteristic polynomial and the eigenvalues of the anti-adjacency matrix. The method used to obtain the general form of the coefficients of the characteristic polynomial is by adding up the determinant values of all directed induced subgraphs, cyclic or acyclic, of the directed cyclic friendship graph. Furthermore, the methods used to find theeigenvalues of the anti-adjacency matrix of directed cyclic friendship graph are the substitution method and factorization method. The result of this researcharethe general form of the coefficients of the characteristic polynomial and the general form of the eigenvalues of the anti-adjacency matrix depend on the number of triangles in directed cyclic friendship graph.

Minimum Total Dominating Energy of Some Standard Graphs

Let G be a simple graph with vertex set V(G) and edge set E(G). A set S of vertices in a graph 𝑮(𝑽,𝑬) is called a total dominating set if every vertex 𝒗 ∈ 𝑽 is adjacent to an element of S. The minimum cardinality of a total dominating set of G is called the total domination number of G which is denoted by 𝜸𝒕 (𝑮). The energy of the graph is defined as the sum of the absolute values of the eigen values of the adjacency matrix. In this paper, we computed minimum total dominating energy of a Friendship Graph, Ladder Graph and Helm graph. The Minimum total dominating energy for bistar graphand sun graph is also determined.

On resolving domination number of friendship graph and its operation

Let G = (V, E) be a simple, finite, and connected graph of order n. A dominating set D ⊆ V(G) such every vertex not in D is adjacent to at least one member of D. A dominating set of smallest size is called a minimum dominating set and it is known as the domination number. The domination number is the minimum cardinality of a dominating set and denoted by γ(G). The other hand, for an ordered set W = {w 1, w 2, w 3, …, wk } of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k − vector r(v|W) = (d(v, w 1), d(v, w 2), d(v, w 3), …, d(v, wk )), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for G is its metric dimension dim(G). A Set of vertices of a graph G that is both resolving and dominating is a resolving dominating set. The minimum cardinality of a resolving dominating set is called resolving domination number γr (G). In this paper, we discussed the resolving domination number of friendship graphs and its operation.

On r-dynamic local irregularity vertex coloring of special graphs

In this paper, we study a new notion of graph coloring, that is a local irregularity vertex r-Dynamic coloring. We combine irregular local and the principles of r-dynamic coloring and we assign color to all vertices by using the weights of local irregularity vertex. We define l : V(G) → {1, 2, …, k} as a vertex irregular k-labeling and w : V(G) → N where w(u) = ∑ v∈N(u) l(v). By a local irregularity vertex coloring, we define a condition for f if for every uv ∈ E(G), w(u) ≠ w(v) and max(l) = min{max{li }; li vertex irregular labeling}. Each vertex of the weight as a color should satisfy the r-dynamic condition, namely |w(N(v)) ≥ min{r, d(v)} and each the adjacent vertices must be different. In this paper, we study the local irregularity vertex r-Dynamic coloring of special graph, namely triangular book graph, central of friendship graph, tapol graph, and rectangular book graph.

On the &lt;i&gt;r&lt;/i&gt;-dynamic coloring of subdivision-edge coronas of a path

This paper deals with the <i>r</i>-dynamic chromatic number of the subdivision-edge corona of a path and exactly one of the following nine types of graphs: a path, a cycle, a wheel, a complete graph, a complete bipartite graph, a star, a double star, a fan graph and a friendship graph.

MULTIPLICATIVE CONNECTIVITY STATUS NEIGHBORHOOD INDICES OF GRAPHS

The connectivity indices are applied to measure the chemical characteristics of compounds in Chemical Graph Theory. In this paper, we introduce the multiplicative atom bond connectivity status neighborhood index, multiplicative geometric-arithmetic status neighborhood index, multiplicative arithmetic-geometric status neighborhood index, multiplicative augmented status neighborhood index of a graph. Also we compute these newly defined indices for some standard graphs, wheel and friendship graphs.

The World of Defacers: Looking Through the Lens of Their Activities on Twitter

Many web-based attacks have been studied to understand how web hackers behave, but web site defacement attacks (malicious content manipulations of victim web sites) and defacers' behaviors have received less attention from researchers. This paper fills this research gap via a computational data-driven analysis of a public database of defacers and defacement attacks and activities of 96 selected defacers who were active on Twitter. We conducted a comprehensive analysis of the data: an analysis of a friendship graph with 10,360 nodes, an analysis on how sentiments of defacers related to attack patterns, and a topical modelling based analysis to study what defacers discussed publicly on Twitter. Our analysis revealed a number of key findings: a modular and hierarchical clustering method can help discover interesting sub-communities of defacers; sentiment analysis can help categorize behaviors of defacers in terms of attack patterns; and topic modelling revealed some focus topics (politics, country-specific topics, and technical discussions) among defacers on Twitter and also geographic links of defacers sharing similar topics. We believe that these findings are useful for a better understanding of defacers' behaviors, which could help design and development of better solutions for detecting defacers and even preventing impeding defacement attacks.

Targeted Sampling from Massive Block Model Graphs with Personalized PageRank

SummaryThe paper provides statistical theory and intuition for personalized PageRank (called ‘PPR’): a popular technique that samples a small community from a massive network. We study a setting where the entire network is expensive to obtain thoroughly or to maintain, but we can start from a seed node of interest and ‘crawl’ the network to find other nodes through their connections. By crawling the graph in a designed way, the PPR vector can be approximated without querying the entire massive graph, making it an alternative to snowball sampling. Using the degree-corrected stochastic block model, we study whether the PPR vector can select nodes that belong to the same block as the seed node. We provide a simple and interpretable form for the PPR vector, highlighting its biases towards high degree nodes outside the target block. We examine a simple adjustment based on node degrees and establish consistency results for PPR clustering that allows for directed graphs. These results are enabled by recent technical advances showing the elementwise convergence of eigenvectors. We illustrate the method with the massive Twitter friendship graph, which we crawl by using the Twitter application programming interface. We find that the adjusted and unadjusted PPR techniques are complementary approaches, where the adjustment makes the results particularly localized around the seed node, and that the bias adjustment greatly benefits from degree regularization.

Total Neighbourhood Prime Labeling of Certain Graphs

In this paper, we investigate the total neighbourhood prime labelling of Lily graphs, Umbrella graph, Friendship graph Fn, Generalised friendship graph and .