Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2018
Volume
Vol. 38
Number
No. 1
Description
Journal Volume
Opuscula Mathematica
Vol. 38 (2018)
Projects
Pages
Articles
Upper bounds for the extended energy of graphs and some extended equienergetic graphs
(Wydawnictwa AGH, 2018) Adiga, Chandrashekar; Rakshith, B. R.
In this paper, we give two upper bounds for the extended energy of a graph one in terms of ordinary energy, maximum degree and minimum degree of a graph, and another bound in terms of forgotten index, inverse degree sum, order of a graph and minimum degree of a graph which improves an upper bound of Das et al. from [On spectral radius and energy of extended adjacency matrix of graphs, Appl. Math. Comput. 296 (2017), 116-123]. We present a pair of extended equienergetic graphs on $n$ vertices for $n\equiv 0(\text{mod } 8)$ starting with a pair of extended equienergetic non regular graphs on $8$ vertices and also we construct a pair of extended equienergetic graphs on $n$ vertices for all $n\geq 9$ starting with a pair of equienergetic regular graphs on $9$ vertices.
The spectrum problem for digraphs of order 4 and size 5
(Wydawnictwa AGH, 2018) Bunge, Ryan C.; DeShong, Steven; El-Zanati, Saad Ibrahim; Fischer, Alexander; Roberts, Dan P.; Teng, Lawrence
The paw graph consists of a triangle with a pendant edge attached to one of the three vertices. We obtain a multigraph by adding exactly one repeated edge to the paw. Now, let $D$ be a directed graph obtained by orientating the edges of that multigraph. For 12 of the 18 possibilities for $D$, we establish necessary and sufficient conditions on $n$ for the existence of a $(K^{*}_{n},D)$-design. Partial results are given for the remaining 6 possibilities for $D$.
Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities
(Wydawnictwa AGH, 2018) Chidouh, Amar; Torres, Delfim F. M.
We prove existence of positive solutions to a boundary value problem depending on discrete fractional operators. Then, corresponding discrete fractional Lyapunov-type inequalities are obtained.
Asymptotic profiles for a class of perturbed Burgers equations in one space dimension
(Wydawnictwa AGH, 2018) Dkhil, Fathi; Hamza, Mohamed Ali; Mannoubi, Bechir
In this article we consider the Burgers equation with some class of perturbations in one space dimension. Using various energy functionals in appropriate weighted Sobolev spaces rewritten in the variables $\frac{\xi}{\sqrt\tau}$ and $\log\tau$, we prove that the large time behavior of solutions is given by the self-similar solutions of the associated Burgers equation.
Wiener index of strong product of graphs
(Wydawnictwa AGH, 2018) Peterin, Iztok; Žigert Pleteršek, Petra
The Wiener index of a connected graph $G$ is the sum of distances between all pairs of vertices of $G$. The strong product is one of the four most investigated graph products. In this paper the general formula for the Wiener index of the strong product of connected graphs is given. The formula can be simplified if both factors are graphs with the constant eccentricity. Consequently, closed formulas for the Wiener index of the strong product of a connected graph $G$ of constant eccentricity with a cycle are derived.