Upper Detour Monophonic Number of a Graph

Titus, P.; Ganesamoorthy, K.

doi:10.1016/j.endm.2016.05.028

Cited by 7 publications

(10 citation statements)

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“…The same result can also be extended so that the detour monophonic number can be prescribed when rad m G < diam m G, and for a proof, one may refer to Ref. [47]. Note 3.16.…”

Section: Theorem 35 No Cut Vertex Of a Connected Graph G Belongs Tomentioning

confidence: 85%

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Monophonic Distance in Graphs

Titus¹,

Santhakumaran²

2018

Graph Theory - Advanced Algorithms and Applications

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For any two vertices u and v in a connected graph G, a u − v path is a monophonic path if it contains no chords, and the monophonic distance d m (u, v) is the length of a longest u − v monophonic path in G. For any vertex v in G, the monophonic eccentricity of v is e m (v) = max {d m (u, v) : u ∈ V}. The subgraph induced by the vertices of G having minimum monophonic eccentricity is the monophonic center of G, and it is proved that every graph is the monophonic center of some graph. Also it is proved that the monophonic center of every connected graph G lies in some block of G. With regard to convexity, this monophonic distance is the basis of some detour monophonic parameters such as detour monophonic number, upper detour monophonic number, forcing detour monophonic number, etc. The concept of detour monophonic sets and detour monophonic numbers by fixing a vertex of a graph would be introduced and discussed. Various interesting results based on these parameters are also discussed in this chapter.Keywords: monophonic path, monophonic distance, detour monophonic number, upper detour monophonic number, forcing detour monophonic number, vertex detour monophonic number, upper vertex detour monophonic number, forcing vertex detour monophonic number Graph Theory -Advanced Algorithms and Applications 116The following result is an easy consequence of the respective definitions. Proposition 2.3. Let u and v be any two vertices in a graph G of order p. ThenResult 2.4. Let u and v be any two vertices in a connected graph G. Then (i) d m (u, v) = 0 if and only if u = v. (ii) d m (u, v) = 1 if and only if uv is an edge of G. (iii) d m (u, v) = p − 1 if and only if G is the path with endvertices u and v.(iv) d(u, v) = d m (u, v) = D(u, v) if and only if G is a tree. Definition 2.5. For any vertex v in a connected graph G, the monophonic eccentricity of v is e m (v) = max {d m (u, v) : u ∈ V}. A vertex u of G such that d m (u, v) = e m (v) is called a monophonic eccentric vertex of v. The monophonic radius and monophonic diameter of G are defined by rad m G = min {e m (v) : v ∈ V} and diam m G = max {e m (v) : v ∈ V}, respectively.Example 2.6. Table 1 shows the monophonic distance between the vertices and also the monophonic eccentricities of vertices of the graph G given in Figure 1. It is to be noted that rad m G = 3 and diam m G = 5.Remark 2.7. In any connected graph, the eccentricity of every two adjacent vertices differs by at most 1. However, this is not true in the case of monophonic distance. For the graph G given in Figure 1, e m (v 5 ) = 3 and e m (v 6 ) = 5. Note 2.8. Any two vertices u and v in a tree T are connected by a unique path, and so d(u, v) = d m (u, v) = D(u, v). Hence rad T = rad m T = rad D T and diam T = diam m T = diam D T.The monophonic radius and the monophonic diameter of some standard graphs are given in Table 2. Figure 1. The graph G in Example 2.2.Monophonic Distance in Graphs http://dx.doi.org/10.5772/intechopen.68668 117Graph Theory -Advanced Algorithms and Applications 132

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“…The same result can also be extended so that the detour monophonic number can be prescribed when rad m G < diam m G, and for a proof, one may refer to Ref. [47]. Note 3.16.…”

Section: Theorem 35 No Cut Vertex Of a Connected Graph G Belongs Tomentioning

confidence: 85%

“…The next theorem is an interesting realization result, and for a proof, one may refer to Ref. [48]. Theorem 3.20.…”

Section: Theorem 35 No Cut Vertex Of a Connected Graph G Belongs Tomentioning

confidence: 93%

Monophonic Distance in Graphs

Titus¹,

Santhakumaran²

2018

Graph Theory - Advanced Algorithms and Applications

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“…A minimal detour monophonic set D of a connected graph G is a subset of V (G) whose any proper subset is not a detour monophonic set of G. The maximum cardinality among all minimal detour monophonic sets is called upper detour monophonic set, denoted by dm + (G) [10].…”

Section: Introductionmentioning

confidence: 99%

Characterization of Upper Detour Monophonic Domination Number

Khayyoom¹

2020

Cubo

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“…the length of a longest u-v monophonic path [1,2,[4][5][6][7] in G. In this present paper, the Gutman index and degree distance index with respect to monophonic distance are defined and studied for some standard graphs like cycle graph and wheel graph.…”

Section: Introductionmentioning

confidence: 99%

Gutman and Degree Monophonic Index of Graphs

Kaladevi¹,

Kavitha²

2019

IJITEE

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A graph with p points and q edges is denoted by G(p,q). An edge joining two non-adjacent points of a path P is called a chord of a path P. A path P is called monophonic if it is a chordless path. For any two points u and v in a connected graph G, the monophonic distance d (u, v) m from u to v is defined as the length of a longest u-v monophonic path in G. The Gutman monophonic index of a graph G is denoted by GutMP(G) and defined by GutMP(G)  d(u)d(v)d (u,v) m and degree monophonic index of G is denoted by DMP(G) and defined by DMP(G) d(u) d(v)d (u,v)    m . The methodology executed in this research paper is to determine the monophonic distance matrix of graphs under consideration. The entries of monophonic distance matrix are calculated by counting the number of edges in the u-v monophonic path. In this paper for some standard graphs, GutMP(G) and DMP(G) are studied which can be applied to derive quantitative structure- property or structure- activity relationships (QSPR / QSAR).

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Upper Detour Monophonic Number of a Graph

Cited by 7 publications

References 4 publications

Monophonic Distance in Graphs

Monophonic Distance in Graphs

Characterization of Upper Detour Monophonic Domination Number

Gutman and Degree Monophonic Index of Graphs

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