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Let G = (V, E) be a finite (non-empty) graph. A monograph is a graph in which all vertices are assigned distinct real number labels so that the positive difference of the end-vertices of every edge is also a vertex label. In this paper we study the properties of monographs and construct signatures for several classes of graph, such as cycles, cycle...
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Let G = (V, E) be a non-empty, finite graph. If the vertices of G can be bijectively labeled by a set S of positive distinct real numbers with two vertices being adjacent if and only if the positive difference of the corresponding labels is in S, then G is called a proper monograph. The set S is called the signature of G and denoted as G(S). Not al...
Let G be a ribbon graph and µ(G) be the number of components of the virtual link formed from G as a cellularly embedded graph via the medial construction. In this paper we first prove that µ(G) ≤ f (G)+γ(G), where f (G) and γ(G) are the number of boundary components and Euler genus of G, respectively. A ribbon graph is said to be extremal if µ(G) =...
Citations
... Sonntag [4] proved that cacti with girth at least 6 are difference graphs, and he conjectured that all cacti are difference graphs. Sugeng and Ryan [5] have provided difference labelings for cycles; fans; cycles with chords; graphs obtained by the one-point union of K n and P m ; and graphs made from any number of copies of a given graph G that has a difference labeling by identifying one vertex of the first with a vertex of the second, a different vertex of the second with the third and so on, In [6], Seoud and Helmi provided a survey of all graphs of order at most 5 and showed that the following graphs are difference graphs: K n for n ≥ 4 with two deleted edges having no vertex in common; K n for n ≥ 6 with three deleted edges having no vertex in common; gear graphs G n for n ≥ 3; P m × P n for m, n ≥ 2; triangular snakes; C 4 -snakes; dragons; graphs consisting of two cycles of the same order joined by an edge, and graphs obtained by identifying the center of a star with a vertex of a cycle. The paper is organized as follows, the next section is devoted to some basic concepts. ...
In this paper, we discuss difference labeling of some standard families of graphs. We prove that Star, Butterfly, Bistar, umbrella and Olive tree are difference graphs. We also introduce difference labelings for some snakes (double triangular snake, irregular triangular snake, alternate $C_n$ snake). Furthermore we introduce a corollary helps us to find a unique difference labeling for the complete graph $K_3$ and all forms of difference labeling for the Star graph. Also this corollary can be used to prove that the complete bipartite graph $K_{2,4}$ is not a difference graph but the proof is very lengthy.
... Sonntag [[5], [6]] investigated the Difference labelling of Cacti and digraphs. Sugeng and Ryan [7] studied the properties of monographs and discovered signatures for cycles, fan graphs, kite graphs and necklaces. Hegde and Vasudeva [4] explored the construction of mod difference digraphs. ...
Let G = (V, E) be a non-empty, finite graph. If the vertices of G can be bijectively labeled by a set S of positive distinct real numbers with two vertices being adjacent if and only if the positive difference of the corresponding labels is in S, then G is called a proper monograph. The set S is called the signature of G and denoted as G(S). Not all proper monographs have the property that a set of idle vertices can be bijectively mapped to the maximum independent sets. As a result, in this paper, we present the proper monograph labelings of several classes of graphs that satisfy the property mentioned above. We present the proper monograph labelings of graphs such as cycles, Cn K 1 , Cycles with paths attached to one or more vertices, and Cycles with an irreducible tree attached to one or more vertices.
... If the signature for a graph contains distinct elements, then the graph is known as a monograph. [14] studied the properties of monographs and discovered signatures for cycles, fan graphs, kite graphs and necklaces. [11] listed signatures for all graphs of order 5 and discovered signatures for gear graphs, triangular snakes, and dragons, among other things. ...
Suppose that G is a simple, vertex-labeled graph and that S is a multiset. Then if there exists a one-to-one mapping between the elements of S and the vertices of G, such that edges in G exist if and only if the absolute difference of the corresponding vertex labels exist in S, then G is an autograph, and S is a signature for G. While it is known that many common families of graphs are autographs, and that infinitely many graphs are not autographs, a non-autograph has never been exhibited. In this paper, we identify the smallest non-autograph: A graph with 6 vertices and 11 edges. Furthermore, we demonstrate that the infinite family of graphs on n vertices consisting of the complement of two non-intersecting cycles contains only non-autographs for n ≥ 8.
... If the signature for a graph contains distinct elements, then the graph is known as a monograph. [14] studied the properties of monographs and discovered signatures for cycles, fan graphs, kite graphs and necklaces. [11] listed signatures for all graphs of order 5 and discovered signatures for gear graphs, triangular snakes, and dragons, among other things. ...
Suppose that $G$ is a simple, vertex-labeled graph and that $S$ is a
multiset. Then if there exists a one-to-one mapping between the elements of $S$
and the vertices of $G$, such that edges in $G$ exist if and only if the
absolute difference of the corresponding vertex labels exist in $S$, then $G$
is an \emph{autograph}, and $S$ is a \emph{signature} for $G$. While it is
known that many common families are graphs are autographs, and that infinitely
many graphs are not autographs, a non-autograph has never been exhibited. In
this paper, we identify the smallest non-autograph: a graph with 6 vertices and
11 edges. Furthermore, we demonstrate that the infinite family of graphs on $n$
vertices consisting of the complement of two non-intersecting cycles contains
only non-autographs for $n \geq 8$.
A graph G is a proper monograph if its vertices can be labeled bijectively by a set S of positive real numbers, called a signature of G, such that two vertices are adjacent if and only if the absolute difference of the corresponding labels is also in S. In this paper, we adapt some concepts from directed graphs, sum graphs and mod difference digraphs to proper monographs to determine their independent sets and vertex coverings by means of their signatures.
