Harishchandra S. Ramane

Karnatak University

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HOSOYA POLYNOMIAL OF THORN TREES, RODS, RINGS, AND STARS

H.B. Walikar Harishchandra S. Ramane Shailaja S. Shirakol Iván Gutman

The Hosoya polynomial is determined for thorn trees, thorn rods, rings, and stars, which are special cases of thorn graphs. By this some earlier results by Bonchev and Klein are generalized. Various distance{based topological indices, namely Wiener index, hyper{Wiener index, Harary index, and reciprocal Wiener index can thus be computed for the classes of graphs under consideration.

Wiener index

Topological index

Reciprocal

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Citations (19)

On the complementary distance energy of join of certain graphs

DOAJ (DOAJ: Directory of Open Access Journals) (2019)

Harishchandra S. Ramane Mahadevappa M. Gundloor

Join (topology)

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Degree Subtraction Adjacency Eigenvalues and Energy of Graphs

Journal of Computer and Mathematical Sciences (2018)

Harishchandra S. Ramane Hemaraddi N. Maraddi Raju B. Jummannaver

Adjacency list

Degree (music)

Subtraction

Adjacency matrix

10.29055/jcms/908

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Citations (1)

Computing the reciprocal distance signless Laplacian eigenvalues and energy of graphs

Le Matematiche (2019)

Abdollah Alhevaz Maryam Baghipur Harishchandra S. Ramane

‎In this paper‎, ‎we study the eigenvalues of the reciprocal distance signless Laplacian matrix of a connected graph and‎ ‎obtain some bounds for the maximum‎ ‎eigenvalue of this matrix‎. ‎We also focus on bipartite graphs and find some bounds for the spectral radius of the reciprocal distance signless Laplacian matrix of this class of graphs‎. ‎Moreover‎, ‎we give bounds for the reciprocal distance signless Laplacian energy.

Reciprocal

Resistance Distance

Spectral Radius

Distance matrix

Connectivity

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Citations (8)

Seidel polynomial and energy of join of two graphs: Seidel polynomial and energy of join of two graphs

Annals of Mathematics and Computer Science (2021)

Harishchandra S. Ramane Daneshwari Patil

For a graph G, Seidel matrix is defined as S(G)=[sij] in which sij=-1 if the vertices vi and vj are adjacent, [sij]=1 if the vertices vi and vj are not adjacent and [sij]=0 for i=j. Seidel polynomial is the polynomial associated to the Seidel matrix. In the present work, we study Seidel polynomial for join of any two graphs by using the concept of coronal of a matrix and hence construct a pair of Seidel equienergetic graphs by joining a regular graph (which is among a pair of Seidel equienergetic graphs of same order and degree) with a non-regular graph.

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On the distance signless Laplacian Estrada index of graphs

Asian-European Journal of Mathematics (2021)

Abdollah Alhevaz Maryam Baghipur Harishchandra S. Ramane Xueliang Li

The distance signless Laplacian eigenvalues [Formula: see text] of a connected graph [Formula: see text] are the eigenvalues of the distance signless Laplacian matrix of [Formula: see text], defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix of vertex transmissions of [Formula: see text]. In this paper, we define and investigate the distance signless Laplacian Estrada index of a graph [Formula: see text] as [Formula: see text], and obtain some upper and lower bounds for [Formula: see text] in terms of other graph invariants. We also obtain some relations between [Formula: see text] and the auxiliary distance signless Laplacian energy of [Formula: see text].

Distance matrix

Diagonal matrix

10.1142/s1793557122500784

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Energy of Graphs

Advances in computer and electrical engineering book series (2019)

Harishchandra S. Ramane

The energy of a graph G is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. The graph energy has close correlation with the total pi-electron energy of molecules calculated with Huckel molecular orbital method in chemistry. A graph whose energy is greater than the energy of complete graph of same order is called hyperenergetic graph. A non-complete graph having energy equal to the energy of complete graph is called borderenergetic graph. Two non-cospectral graphs are said to be equienergetic graphs if they have same energy. In this chapter, the results on graph energy are reported. Various bounds for graph energy and its characterization are summarized. Construction of hyperenergetic, borderenergetic, and equienergetic graphs are reported.

Graph energy

Adjacency matrix

Symmetric graph

Butterfly graph

Regular graph

10.4018/978-1-5225-9380-5.ch011

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Citations (6)

Harary energy of complement of line graphs of regular graphs

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics (2020)

Harishchandra S. Ramane K. Ashoka

The Harary matrix of a graph $G$ is defined as $H(G) = [h_{ij}]$, where $h_{ij} =\frac{1}{d(v_i, v_j)}$, if $i \neq j$ and $h_{ij} = 0$, otherwise, where $d(v_i, v_j)$ is the distance between the vertices $v_i$ and $v_j$ in $G$. The $H$-energy of $G$ is defined as the sum of the absolute values of the eigenvalues of Harary matrix. Two graphs are said to be $H$-equienergetic if they have same $H$-energy. In this paper we obtain the $H$-energy of the complement of line graphs of certian regular graphs interms of the order and regularity of a graph and thus constructs pairs of $H$-equienergetic graphs of same order and having different $H$-eigenvalues.

Complement

10.31801/cfsuasmas.630087

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BEHAVIOR OF ENERGY FOR A CLASS OF CLUSTER GRAPHS

Advances in Mathematics Scientific Journal (2020)

Sumedha Shinde Shaila B. Gudimani Narayan Swamy Harishchandra S. Ramane

10.37418/amsj.9.7.47

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Degree exponent polynomial of graphs obtained by some graph operations

Electronic Notes in Discrete Mathematics (2017)