On the extremal energy of bicyclic digraphs

Applied Mathematics and Computation, 2019

Graph energy can be extended to digraphs via the trace norm. The trace norm of a digraph is the trace norm of its adjacency matrix, i.e. the sum of its singular values. In this work we find the oriented graphs that attain minimal and maximal trace norm over the set of oriented bicyclic graphs.

Proyecciones, 2016

The energy of a digraph is defined as E (D) = n P k=1 |Re (z k)|, where This expression for the energy has many applications in the study of extremal values of the energy in special classes of digraphs. In this paper we consider the set D * (C n) of all strongly connected digraphs whose underlying graph is the cycle C n , and characterize those whose characteristic polynomial is of the form (0.1). As a consequence, we find the extremal values of the energy based on (0.2).

Iranian journal of mathematical chemistry, 2021

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. Let $z_1,ldots,z_n$ be the eigenvalues of an $n$-vertex digraph $D$. Then we give a new notion of energy of digraphs defined by $E_p(D)=sum_{k=1}^{n}|{Re}(z_k) {Im}(z_k)|$, where ${Re}(z_k)$ (respectively, ${Im}(z_k)$) is real (respectively, imaginary) part of $z_k$. We call it $p$-energy of the digraph $D$. We compute $p$-energy formulas for directed cycles. For $ngeq 12$, we show that $p$-energy of directed cycles increases monotonically with respect to their order. We find unicyclic digraphs with smallest and largest $p$-energy. We give counter examples to show that the $p$-energy of digraph does not possess increasing--property with respect to quasi-order relation over the set $mathcal{D}_{n,h}$, where $mathcal{D}_{n,h}$ is the set of $n$-vertex digraphs with cycles of length $h$. We find the upper bound for $p$-energy and give all those digraphs which attain this bound. Moreover, we construct few families...

Asian-European Journal of Mathematics, 2018

Iota energy of signed digraphs is defined as sum of absolute values of imaginary parts of its eigenvalues. Extremal energy and iota energy of unicyclic signed digraphs is known. In this paper, we address the problem of finding signed digraphs with extremal iota energy among vertex-disjoint bicyclic signed digraphs of fixed order.

2021

The energy of a sidigraph is defined as the sum of absolute values of real parts of its eigenvalues. The iota energy of a sidigraph is defined as the sum of absolute values of imaginary parts of its eigenvalues. Recently a new notion of energy of digraphs is introduced which is called the total energy of digraphs. In this paper, we extend this concept of total energy to sidigraphs. We compute total energy formulas for negative directed cycles and show that the total energy of negative directed cycles with fixed order increases monotonically. We introduce complex adjacency matrix to give the integral representation for total energy of sidigraphs. We discuss the increasing property of total energy over some particular subfamilies of Sn,h, where Sn,h contains n-vertex sidigraphs with each cycle having length h. Using the CauchySchwarz inequality, we find upper bound for the total energy of sidigraphs. Finally, we find the class of noncospectral equienergetic sidigraphs.

The energy of a digraph D with eigenvalues z 1 , z 2 ,. .. , zn is defined as E(D) = n ∑ j=1 |ℜz j |, where ℜz j is the real part of the complex number z j. In this paper, we characterize some positive reals that cannot be the energy of a digraph. We also obtain a sharp lower bound for the energy of strongly connected digraphs.

Linear Algebra and its Applications, 2012

Let D be a digraph with vertex set V (D) and A be the adjacency matrix of D. In this paper, we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among strongly connected bicyclic digraphs. Furthermore, we show that any strongly connected bicyclic digraph is determined by the spectrum.

Acta Universitatis Sapientiae, Informatica, 2021

Let S = (G, σ) be a signed graph of order n and size m and let x1, x2, ..., xn be the eigenvalues of S. The energy of S is defined as ɛ ( S ) = ∑ j = 1 n | x j | \varepsilon \left( S \right) = \sum\limits_{j = 1}^n {\left| {{x_j}} \right|} . A connected signed graph is said to be bicyclic if m=n + 1. In this paper, we determine the bicyclic signed graphs with first 20 minimal energies for all n ≥ 30 and with first 16 minimal energies for all 17 ≤ n ≤ 29.

Linear and Multilinear Algebra, 2016

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. If D is a digraph of order n with eigenvalues z 1 , z 2 ,. .. , z n , then its energy is defined by E(D) = n k=1 Re(z k). We give a new notion of energy of digraphs defined by E c (D) = n k=1 Im(z k) and call it the iota energy of the digraph D. It is shown that the Coulson's integral formula remains valid for iota energy. We also find the unicyclic digraphs with extremal iota energies among the class of unicyclic digraphs with a fixed order. Furthermore, it is shown that the iota energy is increasing over the set D n,h of n-vertex digraphs with cycles of length h, with respect to a quasi-order relation. We also generalize the increasing property of the energy over the set D n,h with respect to quasi-order relation.

Matrix Science Mathematic, 2018

The iota energy of an n-vertex digraph D is defined by Ec () = ∑ |Im(k)| =1 , where z1,. . ., zn are eigenvalues of D and Im(zk) is the imaginary part of eigenvalue zk. The iota energy of an n-vertex sidigraph can be defined analogously. In this paper, we define a class Fn of n-vertex tricyclic digraphs containing five linear subdigraphs such that one of the directed cycles does not share any vertex with the other two directed cycles and the remaining two directed cycles are of same length sharing at least one vertex. We find the digraphs in Fn with minimal and maximal iota energy. We also consider a similar class of tricyclic sidigraphs and find extremal values of iota energy among the sidigraphs in this class.