Energy of signed digraphs

Abstract

In this paper we extend the concept of energy to signed digraphs. We obtain Coulson's integral formula for energy of signed digraphs. Formulae for energies of signed directed cycles are computed and it is shown that energy of non cycle balanced signed directed cycles increases monotonically with respect to number of vertices. Characterization of signed digraphs having energy equal to zero is given. We extend the concept of non complete extended p sum (or briefly, NEPS) to signed digraphs. An infinite family of equienergetic signed digraphs is constructed. Moreover, we extend McClelland's inequality to signed digraphs and also obtain sharp upper bound for energy of signed digraph in terms the number of arcs. Some open problems are also given at the end.2010 Mathematics Subject Classification. 05C50, 05C22,05C76.

Coxeter energy of graphs

“…[19,53]) in the sense of Simson [57]. Coxeter matrices and their spectra defined for bigraphs and posets are studied in [3,15,17,18,33,45,55,[57][58][59] in context of applications in Gram's classifications of integral quadratic and bilinear forms, problems of Diophantine geometry, computational number theory, representation theory of groups, algebras and posets and spectral graph theory.…”

“…. , μ n ∈ C are all eigenvalues of the adjacency matrix Ad D of D as in Definition 2.1(a), see also [39,53], cf. Introduction.…”

“…We refer also to [52] (resp. [19,53]), where the authors generalize the definition of energy for digraphs (resp. signed graphs, signed digraphs).…”

Seidel Equienergetic Graphs

“…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.…”

Extremal energy of digraphs with two linear subdigraphs

“…The authors derive an upper bound for the skew energy of digraph and computed skew energies of directed cycles. Pirzada and Bhat [10] extended the concept of energy of digraphs to the energy of signed digraphs. The authors extended the notion of NEPS to signed digraphs and constructed infinite families of non cospectral equienergetic signed digraphs.…”