اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSome inequalities for orderings of acyclic digraphs
260
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Let $D=(V,A)$ be an acyclic digraph. For $xin V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V rightarrow [1,|V|] $ that has the property that for all $x,yin V$ if $(x,y)in A$, then $g(x) < g(y)$. We prove that for every acyclic ordering $g$ of $D$ the following inequality holds: [sum_{xin V} e_{_{D}}(x)cdot g(x) ~geq~ frac{1}{2} sum_{xin V}[e_{_{D}}(x)]^2~.] The class of acyclic digraphs for which equality holds is determined as the class of comparbility digraphs of posets of order dimension two.
قيم البحث
اقرأ أيضاً
The concept of energy of a signed digraph is extended to iota energy of a signed digraph. The energy of a signed digraph $S$ is defined by $E(S)=sum_{k=1}^n|text{Re}(z_k)|$, where $text{Re}(z_k)$ is the real part of eigenvalue $z_k$ and $z_k$ is the
eigenvalue of the adjacency matrix of $S$ with $n$ vertices, $k=1,2,ldots,n$. Then the iota energy of $S$ is defined by $E(S)=sum_{k=1}^n|text{Im}(z_k)|$, where $text{Im}(z_k)$ is the imaginary part of eigenvalue $z_k$. In this paper, we consider a special graph class for bicyclic signed digraphs $mathcal{S}_n$ with $n$ vertices which have two vertex-disjoint signed directed even cycles. We give two iota energy orderings of bicyclic signed digraphs, one is including two positive or two negative directed even cycles, the other is including one positive and one negative directed even cycles.
In this paper, we characterize the extremal digraphs with the maximal or minimal $alpha$-spectral radius among some digraph classes such as rose digraphs, generalized theta digraphs and tri-ring digraphs with given size $m$. These digraph classes are
denoted by $mathcal{R}_{m}^k$, $widetilde{boldsymbol{Theta}}_k(m)$ and $INF(m)$ respectively. The main results about spectral extremal digraph by Guo and Liu in cite{MR2954483} and Li and Wang in cite{MR3777498} are generalized to $alpha$-spectral graph theory. As a by-product of our main results, an open problem in cite{MR3777498} is answered. Furthermore, we determine the digraphs with the first three minimal $alpha$-spectral radius among all strongly connected digraphs. Meanwhile, we determine the unique digraph with the fourth minimal $alpha$-spectral radius among all strongly connected digraphs for $0le alpha le frac{1}{2}$.
Visibility representation of digraphs was introduced by Axenovich, Beveridge, Hutch-inson, and West (emph{SIAM J. Discrete Math.} {bf 27}(3) (2013) 1429--1449) as a natural generalization of $t$-bar visibility representation of undirected graphs. A {
it $t$-bar visibility representation} of a digraph $G$ assigns each vertex at most $t$ horizontal bars in the plane so that there is an arc $xy$ in the digraph if and only if some bar for $x$ sees some bar for $y$ above it along an unblocked vertical strip with positive width. The {it visibility number} $b(G)$ is the least $t$ such that $G$ has a $t$-bar visibility representation. In this paper, we solve several problems about $b(G)$ posed by Axenovich et al. and prove that determining whether the bar visibility number of a digraph is $2$ is NP-complete.
In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of pa
ths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.
Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $kgeq2$. This enables us to extend the $k$-colored partition function multiplicatively to a function o
n $k$-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
