Papers by Vinayak V Manjalapur
International Journal of Mathematical Archive, 2015
The eccentricity of a vertex v of graph G is the largest distance between and any other vertex of... more The eccentricity of a vertex v of graph G is the largest distance between and any other vertex of a graph . The reciprocal complementary Wiener (RCW) index of is defined as, , where D is the diameter of G and is the distance between the vertices and . In this paper we have obtained bounds for the index in terms of eccentricities and given an algorithm to compute the index.
Mathematics
For a (molecular) graph G, the extended adjacency index E A ( G ) is defined as equation . In thi... more For a (molecular) graph G, the extended adjacency index E A ( G ) is defined as equation . In this paper we introduce some graph transformations which increase or decrease the extended adjacency ( E A ) index. Also, we obtain the extremal acyclic, unicyclic and bicyclic graphs with minimum and maximum of the E A index by a unified method, respectively.
— Harary index of graph G is defined as the sum of reciprocal of distance between all pairs of ve... more — Harary index of graph G is defined as the sum of reciprocal of distance between all pairs of vertices of the graph G and is denoted by () H G. Eccentricity of vertex v in G is the distance to a vertex farthest from v. In this paper we obtain some bounds for () H G in terms of eccentricities. Further we extend these results to the self-centered graphs and also we have given simple algorithm to find the Harary index of graphs.
Theorem 7 in [H. B. Walikar, V. S. Shigehalli, H. S. Ramane, Bounds on the Wiener number of a gra... more Theorem 7 in [H. B. Walikar, V. S. Shigehalli, H. S. Ramane, Bounds on the Wiener number of a graph, MATCH Commun. Math. Comput. Chem., 50 (2004), 117–132] related to the Wiener number of a graph interms of the chromatic number is not correct. This note provides the correct version of the theorem.
The Harary matrix or reciprocal distance matrix of a graph í µí°ºí µí°º is defined as í µí±
í µí±... more The Harary matrix or reciprocal distance matrix of a graph í µí°ºí µí°º is defined as í µí±
í µí±
í µí±
í µí±
(í µí°ºí µí°º) = [í µí±í µí± í µí±í µí± í µí±í µí± ] , in which í µí±í µí± í µí±í µí± í µí±í µí± = 1 í µí±í µí± í µí±í µí± í µí±í µí± if í µí±í µí± ≠ í µí±í µí± and í µí±í µí± í µí±í µí± í µí±í µí± = 0 if í µí±í µí± = í µí±í µí±, where í µí±í µí± í µí±í µí± í µí±í µí± is the distance between the í µí±í µí± í µí±¡í µí±¡ℎ and í µí±í µí± í µí±¡í µí±¡ℎ vertex of í µí°ºí µí°º. The Harary energy í µí°»í µí°»í µí°»í µí°»(í µí°ºí µí°º) of í µí°ºí µí°º is defined as the sum of the absolute values of the eigenvalue of the Harary matrix of graph í µí°ºí µí°º. Two graphs í µí°ºí µí°º 1 and í µí°ºí µí°º 2 are said to be Harary equienergetic if í µí°»í µí°»í µí°»í µí°»(í µí°ºí µí°º 1) = í µí°»í µí°»í µí°»í µí°»(í µí°ºí µí°º 2). In this paper we obtain the Harary eigenvalues and Harary energy of the join of regular graphs of diameter less than or equal to two and thus construct the Harary equienergetic graphs on í µí±í µí± vertices, for all í µí±í µí± ≥ 6 having different Harary eigenvalues.
Let G be a graph on n vertices and B0(G) be its edge-vertex incidence matrix. The
row in B0(G) co... more Let G be a graph on n vertices and B0(G) be its edge-vertex incidence matrix. The
row in B0(G) corresponding to an edge e, denoted by s(e) is a string which belongs
to Zn2
, a set of n-tuples over a field of order two. The Hamming distance between
the strings s(e) and s(f) is the number of positions in which s(e) and s(f) differ.
Hamming index of a graph is the sum of Hamming distances between all pairs of
strings. In this paper we obtain the Hamming index of graphs generated by an
edge-vertex incidence matrix along with an algorithm.
Abstract: Let G be a connected graph with vertex set V(G) = {v1,v2, . . . ,vn} . The
distance bet... more Abstract: Let G be a connected graph with vertex set V(G) = {v1,v2, . . . ,vn} . The
distance between two vertices vi and vj , denoted by d(vi,vj) is the length of a
shortest path joining vi and vj . Reciprocal Wiener index of a graph G is defined as
RW(G) = Âi<j
1
d(vi,vj) . Reciprocal complementary Wiener index of a graph G is defined
as RCW(G) = Âi<j
1
1+D−dij , where dij is the distance between two vertices vi and vj
and D is the diameter of G . In this paper we obtain bounds for the reciprocal Wiener
index and reciprocal complementaryWiener index of line graphs.
Drafts by Vinayak V Manjalapur
In this paper, we obtain some lower and upper bounds for the maximum eigenvalue of the complement... more In this paper, we obtain some lower and upper bounds for the maximum eigenvalue of the complementary distance signless Laplacian matrix (CDL +) of a graph G. We have given a Nordhaus-Gaddum type results for the spectral radius (ρ1) of CDL + matrix of a graph G. Also, we have consider bipartite graphs and find some bounds for the ρ1 of the CDL + matrix of this class of graphs. Moreover, we give bounds for the complementary distance signless Laplacian energy.
The general sum-connectivity index, general product-connectivity index, general Zagreb index and ... more The general sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graphs of subdivison graphs of tadpole graphs, wheels and ladders have been reported in the literature. In this paper , we obtain general expressions for these topological indices for the line graph of the subdivison graphs, thus generalizing the existing results.
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Papers by Vinayak V Manjalapur
row in B0(G) corresponding to an edge e, denoted by s(e) is a string which belongs
to Zn2
, a set of n-tuples over a field of order two. The Hamming distance between
the strings s(e) and s(f) is the number of positions in which s(e) and s(f) differ.
Hamming index of a graph is the sum of Hamming distances between all pairs of
strings. In this paper we obtain the Hamming index of graphs generated by an
edge-vertex incidence matrix along with an algorithm.
distance between two vertices vi and vj , denoted by d(vi,vj) is the length of a
shortest path joining vi and vj . Reciprocal Wiener index of a graph G is defined as
RW(G) = Âi<j
1
d(vi,vj) . Reciprocal complementary Wiener index of a graph G is defined
as RCW(G) = Âi<j
1
1+D−dij , where dij is the distance between two vertices vi and vj
and D is the diameter of G . In this paper we obtain bounds for the reciprocal Wiener
index and reciprocal complementaryWiener index of line graphs.
Drafts by Vinayak V Manjalapur
row in B0(G) corresponding to an edge e, denoted by s(e) is a string which belongs
to Zn2
, a set of n-tuples over a field of order two. The Hamming distance between
the strings s(e) and s(f) is the number of positions in which s(e) and s(f) differ.
Hamming index of a graph is the sum of Hamming distances between all pairs of
strings. In this paper we obtain the Hamming index of graphs generated by an
edge-vertex incidence matrix along with an algorithm.
distance between two vertices vi and vj , denoted by d(vi,vj) is the length of a
shortest path joining vi and vj . Reciprocal Wiener index of a graph G is defined as
RW(G) = Âi<j
1
d(vi,vj) . Reciprocal complementary Wiener index of a graph G is defined
as RCW(G) = Âi<j
1
1+D−dij , where dij is the distance between two vertices vi and vj
and D is the diameter of G . In this paper we obtain bounds for the reciprocal Wiener
index and reciprocal complementaryWiener index of line graphs.