Given a graph F , a graph G is uniquely F -saturated if F is not a subgraph of G and adding any edge of the complement to G completes exactly one copy of F . In this paper we study uniquely Ct-saturated graphs. We prove the following: (1) a graph is uniquely C5-saturated if and only if it is a friendship graph. (2) There are no uniquely C6-saturated graphs or uniquely C7-saturated graphs. (3) F...

The friendship paradox is a sociological phenomenon stating that most people have fewer friends than their friends do. The generalized friendship paradox refers to the same observation for attributes other than degree, and it has been observed in Twitter and scientific collaboration networks. This paper takes an analytical approach to model this phenomenon. We consider a preferential attachment...

The friendship paradox is revisited by considering both local and global averages of friends. How the economics of attention affects the recruitment of friends is examined. Statistical implications of varying individual attentions are investigated and it is argued that this is one reason why the mean of friends is higher than the median in social networks. The distribution of friends skews to t...

Our friends have more friends than we do. That is the basis of the friendship paradox. In mathematical terms, the mean number of friends of friends is higher than the mean number of friends. In the present study, we analyzed the relationship between the mean degree of vertices (individuals), 〈k〉, and the mean number of friends of friends, 〈kFF 〉, in scale-free networks with degrees ranging from...

In this paper we provide a purely combinatorial proof of the Friendship Theorem, which has been first proven by P. Erdös et al. by using also algebraic methods. Moreover, we generalize this theorem in a natural way, assuming that every pair of nodes occupies ≥ 2 common neighbors. We prove that every graph, which satisfies this generalized -friendship condition, is a regular graph.

A graph G is H-saturated if G does not contain H as a subgraph but for any nonadjacent vertices u and v, G+uv containsH as a subgraph. The parameter sat(H, n) is theminimum number of edges in an H-saturated graph of order n. In this paper, we determine sat(H, n) for sufficiently large nwhen H is a union of cliques of the same order, an arbitrary union of two cliques and a generalized friendship...

the purpose of this study is to investigate the relation between identity styles and commitment with quality of friendship among the students in third grade of high school. 400 participants from nineteen educational districts were chosen randomly. three quesnonnaires were used: identity style inventory, the quality of friendship, the reciprocity of friendship . the results indicated that inform...

The Friendship theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let λ be any nonnegative integer and μ be any positive integer. Suppose each pair of friends have exactly λ common friends and each pair of strangers have exactly μ common friends in a pa...

The friendship paradox refers to the sociological observation that, while the people’s assessment of their own popularity is typically self-aggrandizing, in reality they are less popular than their friends. The generalized friendship paradox is the average alter superiority observed empirically in social settings, scientific collaboration networks, as well as online social media. We posit a qua...

objective: one of the main cultural issues in universities is relationships between male and female students. due to the adverse consequences of this issue, explaining of the beyond the causes of friendship with opposite-sex, is one of the first steps in planning for youth issues. methods: this qualitative research was conducted based on content analysis. nineteen female university students wer...