Minimum Covering Harary Energy of a Graph

Kanna, M.R.R.; Dharmendra, B. N.; Kumar, Rajesh

doi:10.12732/ijpam.v90i3.10

Cited by 1 publication

(4 citation statements)

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“…Example 6 Complete graph K n . For n ≥ 2, the eigenvalues of the minimum covering Harary matrix of complete graph K n was determined in [1,22] as Spec(…”

Section: Mcrdsl Energy Of Some Standard Graphsmentioning

confidence: 99%

“…For a vertex v, deg(v) denotes the degree of v. Energy of a graph introduced by Ivan Gutman [12] as the sum of the absolute values of the eigenvalues of adjacency matrix of G. The concept of energy of graph have been extensively studied; for more information we refer to surveys [13,23,24]. Various kinds of graph energy such as Laplacian energy [14], minimum covering energy [1], minimum covering distance energy [21], and minimum covering Harary energy [22] of a graph were proposed and some mathematical aspects of them were investigated. A subset C of V(G) is called a vertex covering set of G if every edge of G is incident to at least one vertex of C. A vertex covering set with minimum cardinality is called minimum vertex covering set.…”

Section: Introductionmentioning

confidence: 99%

“…The minimum covering energy of G is defined to be absolute values of the eigenvalues of A C (G). The minimum covering Harary matrix [22] of G, is a square matrix n × n defined as RDC(G) = (RDC ij ), where…”

Section: Introductionmentioning

confidence: 99%

“…, λ n are eigenvalues of H C (G). The mathematical aspects of the minimum covering Harary energy was reported in [22].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Minimum covering reciprocal distance signless Laplacian energy of graphs

Alhevaz

Hashemi

Alizadeh

2018

Acta Universitatis Sapientiae, Informatica

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Let G be a simple connected graph. The reciprocal transmission Tr′G(ν) of a vertex ν is defined as $${\rm{Tr}}_{\rm{G}}^\prime ({\rm{\nu }}) = \sum\limits_{{\rm{u}} \in {\rm{V}}(G)} {{1 \over {{{\rm{d}}_{\rm{G}}}(u,{\rm{\nu }})}}{\rm{u}} \ne {\rm{\nu }}.} $$ The reciprocal distance signless Laplacian (briefly RDSL) matrix of a connected graph G is defined as RQ(G)= diag(Tr′ (G)) + RD(G), where RD(G) is the Harary matrix (reciprocal distance matrix) of G and diag(Tr′ (G)) is the diagonal matrix of the vertex reciprocal transmissions in G. In this paper, we investigate the RDSL spectrum of some classes of graphs that are arisen from graph operations such as cartesian product, extended double cover product and InduBala product. We introduce minimum covering reciprocal distance signless Laplacian matrix (or briey MCRDSL matrix) of G as the square matrix of order n, RQC(G) := (qi;j), $${{\rm{q}}_{{\rm{ij}}}} = \left\{ {\matrix{ {1 + {\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \in {\rm{C}}} \cr {{\rm{Tr}}\prime ({{\rm{\nu }}_{\rm{i}}})} & {{\rm{if}}} & {{\rm{i = j}}} & {{\rm{and}}} & {{{\rm{\nu }}_{\rm{i}}} \notin {\rm{C}}} \cr {{1 \over {{\rm{d(}}{{\rm{\nu }}_{\rm{i}}},{{\rm{\nu }}_{\rm{j}}})}}} & {{\rm{otherwise}}} & {} & {} & {} \cr } } \right.$$ where C is a minimum vertex cover set of G. MCRDSL energy of a graph G is defined as sum of eigenvalues of RQC. Extremal graphs with respect to MCRDSL energy of graph are characterized. We also obtain some bounds on MCRDSL energy of a graph and MCRDSL spectral radius of 𝒢, which is the largest eigenvalue of the matrix RQC (G) of graphs.

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“…Example 6 Complete graph K n . For n ≥ 2, the eigenvalues of the minimum covering Harary matrix of complete graph K n was determined in [1,22] as Spec(…”

Section: Mcrdsl Energy Of Some Standard Graphsmentioning

confidence: 99%