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A set of vertices is $k$-sparse if it induces a graph with a maximum degree of at most $k$. In this missive, we consider the order of the largest $k$-sparse set in a triangle-free graph of fixed order. We show, for example, that every triangle-free graph of order 11 contains a 1-sparse 5-set; every triangle-free graph of order 13 contains a 2-spars...
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Given a family \(\mathcal {F}\) of graphs, we say that a graph G is \(\mathcal {F}\)-saturated if G does not contain any member of \(\mathcal {F}\), but for any edge \(e\in E(\overline{G})\) the graph \(G+e\) does contain a member of \(\mathcal {F}\). The \(\mathcal {F}\)-saturation game is played by two players starting with an empty graph and add...
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In a closed connected orientable 3-manifold associated with an orientation-preserving smooth finite group action, we construct setwise invariant hyperbolic spatial graphs with given singularity. As an application, we provide a condition under which symmetries of abstract graphs are realizable by symmetries of the 3-sphere through hyperbolic spatial...
Erasures are a common problem that arises while signals or data are being transmitted. A profound challenge in frame theory is to find the optimal dual frames ($OD$-frames) to minimize the reconstruction error if erasures occur. In this paper, we study the optimal duals of frames generated by graphs. First, we characterize walk-regular graphs. Then...
Citations
... The idea of associating graphs with groups originated in Kelly graphs and has developed into an important research focus in modern algebraic graph theory A notable development in recent years has been to identify and study a graph type defined in categories, such as power graphs [1,7], augmented power maps [2,3], navigation diagrams [9,10], non-navigation diagrams [1, 4 ,5], subgroup inclusion statistics [8], and so on. Many works such as [6] have been done on these topics. For a comprehensive description of group-defined records, [10] is a good reference. ...
This paper explores Subgroup Product Graphs (SPG) in cyclic groups, presenting a Vertex Degrees Formula based on the prime factorization of a positive integer n. The Isolated Vertex Property asserts that for a positive integer n, the SPG γ_sp(G) lacks isolated vertices. The Matrix Degree and Edge Formula provide a matrix representation and calculate the edges in SPG. Additionally, a Subgraph Relation identifies the complete graph K π( n ) as a subgraph in γ_sp(G). Specific Examples illustrate vertex degrees for different n values. In essence, the study contributes isomorphisms, characterizes properties, and computes degrees and edges for diverse subgroups in Subgroup Product Graphs.
Let G be a finite group with normal degree, cyclic degree and Sylow degree
Nor(G), Cyc(G) and Syl(G) respectively. The normality degree, cyclicity
degree and Sylow degree of G is the probability that two elements randomly
chosen normal subgroup, cyclic subgroup or Sylow subgroup in G denoted as
PNor(G),PCyc(G) and PSyl(G) respectively. This concept is used to determine
the normal degree, cyclic degree and Sylow degree of a group. The probability
can be obtained by finding a normality degree, cyclicity degree and Sylow
degree element table and conjugacy classes. The concept of parameters has
been generalized to Cn and D2n groups, one of these generalizations is the
probability that a group element is not subgroup in G.
