Distance spectra and distance energies of iterated line graphs of regular graphs

Ramane, Harishchandra S.; Revankar, D. S.; Gutman, Ivan; Walikar, H. B.

doi:10.2298/pim0999039r

Cited by 36 publications

(9 citation statements)

References 10 publications

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“…Indulal et al [17] constructed pairs of D-equienergetic graphs on n vertices for n ≡ 1 (mod 3) and for n ≡ 0 (mod 6). Ramane et al [18] proved that if G 1 and G 2 are r-regular graphs on n vertices and diam(G i ) 2 , i = 1, 2, then…”

Section: Introductionmentioning

confidence: 99%

Distance spectra and distance energy of integral circulant graphs

Ilić

2010

Linear Algebra and its Applications

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The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix D. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs (G 1 , G 2 ) of integral circulant graphs with equal distance energy -in the first family G 1 is subgraph of G 2 , while in the second family the diameter of both graphs is three.

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Section: Introductionmentioning

confidence: 99%

Distance spectra and distance energy of integral circulant graphs

Ilić

2010

Linear Algebra and its Applications

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“…Metric dimension has been deeply elaborated in surveys [6,7]. The line graph L(G) of a graph G is defined as, the graph whose vertices are the edges of G, with two adjacent vertices if the corresponding edges share the common vertex in G. Also, metric properties of line graphs were studied to a great extent in [8][9][10][11][12][13][14][15][16][17][18]. The line graph of a graph G is helpful to find edge distances using the same technique of finding vertex distances of the graph G.…”

Section: Introductionmentioning

confidence: 99%

Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph

et al. 2018

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“…For obvious reason Seidel cospectral graphs are Seidel equienergetic. Several results on equienergetic graphs [2,4,5,15,24,27,28,29,30,31], distance equienergetic graphs [14,23,26], Laplacian equienergetic graphs [8], equienergetic digraphs [17], skew equienergetic digraphs [1,25], equienergetic sigraphs [3,9,10,19] and equienergetic sidigraphs [21] were reported in the literature.…”

Section: Introductionmentioning

confidence: 99%

Seidel Equienergetic Graphs

Ramane

Gundloor

Hosamani³

2016

BMSA

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Abstract. The Seidel matrix S(G) of a graph G is the square matrix with diagonal entries zeroes and off diagonal entries are -1 or 1 corresponding to the adjacency and non-adjacency. The Seidel energy SE(G) of G is defined as the sum of the absolute values of the eigenvalues of S(G). Two graphs G 1 and G 2 are said to be Seidel equienergetic if SE(G 1 ) = SE(G 2 ). We establish an expression for the characteristic polynomial of the Seidel matrix and for the Seidel energy of the join of regular graphs. Thereby construct Seidel non cospectral, Seidel equienergetic graphs on n vertices, for all n ≥ 12.

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Distance spectra and distance energies of iterated line graphs of regular graphs

Abstract: Abstract. The distance or D-eigenvalues of a graph G are the eigenvalues of its distance matrix. The distance or D-energy E D (G) of the graph G is the sum of the absolute values of its D-eigenvalues. Two graphs G 1 and G 2 are said to be

Cited by 36 publications

References 10 publications

Distance spectra and distance energy of integral circulant graphs

Distance spectra and distance energy of integral circulant graphs

Edge Version of Metric Dimension and Doubly Resolving Sets of the Necklace Graph

Seidel Equienergetic Graphs

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