Sep 2026· Journal of Interconnection Networks (JOIN)· 0 citations
Abstract
The eccentricity of any vertex [Formula: see text] in a connected graph [Formula: see text] is the length of the largest distance from [Formula: see text] to any other vertex in [Formula: see text]. The eccentric graph of any graph [Formula: see text], denoted by [Formula: see text], is a graph with the same vertex set as [Formula: see text] and two vertices in [Formula: see text] are adjacent if the distance between those vertices is equal to the eccentricity of either of the vertices. A graph [Formula: see text] is properly connected if there is a properly colored path between every pair of vertices in it. In this paper, we define the proper eccentric graph for any properly connected graph [Formula: see text]. We examine the connectivity, proper eccentricity, proper diameter, and several other graph invariants of a proper eccentric graph of the join of any two connected graphs.
Let [Formula: see text] be a graph with no isolated vertex. A dominated coloring of [Formula: see text] is a proper coloring of [Formula: see text] such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of [Formula: see text] is called the dominated...
A 2-distance [Formula: see text]-coloring of a graph is a proper coloring of the vertices of the graph using [Formula: see text] colors such that any two vertices at distance two or less get distinct colors. The 2-distance chromatic number of a graph [Formula: see text], denoted as [Formula: see text], is the minimum i...
Sara Al Hajjar· Discrete Mathematics, Algori...· 0 citations
Let [Formula: see text] be a ring with nonzero identity and [Formula: see text] denotes the set of idempotents in [Formula: see text]. A graph of [Formula: see text] with respect to idempotents [Formula: see text] is a graph whose vertices are elements of [Formula: see text] and two vertices [Formula: see text] and [Fo...
Dipika B. Patil, Avinash Patil· Journal of Algebra and its A...· 0 citations
A [Formula: see text]-power hypergraph is obtained from a graph by adding [Formula: see text] new vertices to each edge of the graph. Let [Formula: see text] be a graph obtained by adding [Formula: see text] edges and [Formula: see text] edges to the two pendant vertices of a path of length 2, respectively. A [Formula:...
A [Formula: see text]-distance [Formula: see text]-coloring of a graph is a coloring of the vertices with [Formula: see text] colors in which any two vertices at distance at most [Formula: see text] receive distinct colors. The [Formula: see text]-distance chromatic number of [Formula: see text], denoted by [Formula: s...
Sara Al Hajjar· Discrete Mathematics, Algori...· 0 citations
A hereditary class [Formula: see text] of graphs is [Formula: see text]-bounded if there is a [Formula: see text]-binding function, say [Formula: see text], such that [Formula: see text], for every [Formula: see text], where [Formula: see text] denotes the chromatic (clique) number of [Formula: see text]. A [Formula: s...
Li Zhang, Xia Hong· Discrete Mathematics, Algori...· 0 citations
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