Abstract
An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is considered in this paper. It is well known that this graph parameter is always nonnegative and represents some measure of deviation of a graph from its regularity. In the last two decades, many papers have been devoted to the study of this parameter. In particular, its lower bound depending on the graph order and diameter was obtained in 2007 by mathematician S. M. Cioabă. In 2017 when studying the upper and the lower bounds of this parameter, M. R. Oboudi made a conjecture that the lower bound of a given parameter for an arbitrary graph is the difference between a maximum degree and a spectral radius of a chain. This is very similar to the analogous statement for the spectral radius of an arbitrary graph whose lower boundary is also the spectral radius of a chain. In this paper, the above conjecture is confirmed for some graph classes.
Highlights
An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is con sidered in this paper
It is well known that this graph parameter is always nonnegative and represents some measure of de viation of a graph from its regularity
Its lower bound depending on the graph order and diameter was obtained in 2007 by mathematician S
Summary
An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is con sidered in this paper. R. Oboudi made a conjecture that the lower bound of a given parameter for an arbitrary graph is the difference between a maximum degree and a spectral radius of a chain. В той же статье им была высказана следующая гипотеза: пусть G – нерегулярный связный граф порядка n, отличный от цепи Pn, тогда справедливо неравенство β(G) > β(Pn).
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