ﻻ يوجد ملخص باللغة العربية
This paper presents sufficient conditions for Hamiltonian paths and cycles in graphs. Letting $lambdaleft( Gright) $ denote the spectral radius of the adjacency matrix of a graph $G,$ the main results of the paper are: (1) Let $kgeq1,$ $ngeq k^{3}/2+k+4,$ and let $G$ be a graph of order $n$, with minimum degree $deltaleft( Gright) geq k.$ If [ lambdaleft( Gright) geq n-k-1, ] then $G$ has a Hamiltonian cycle, unless $G=K_{1}vee(K_{n-k-1}+K_{k})$ or $G=K_{k}vee(K_{n-2k}+overline{K}_{k})$. (2) Let $kgeq1,$ $ngeq k^{3}/2+k^{2}/2+k+5,$ and let $G$ be a graph of order $n$, with minimum degree $deltaleft( Gright) geq k.$ If [ lambdaleft( Gright) geq n-k-2, ] then $G$ has a Hamiltonian path, unless $G=K_{k}vee(K_{n-2k-1}+overline {K}_{k+1})$ or $G=K_{n-k-1}+K_{k+1}$ In addition, it is shown that in the above statements, the bounds on $n$ are tight within an additive term not exceeding $2$.
Let $G$ be a simple graph with maximum degree $Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Delta(G)lfloor |V(H)|/2 rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $Delta(G)>|V(G)|/3$ has chromatic index $Delta(G)$ i
In this paper, we present a spectral sufficient condition for a graph to be Hamilton-connected in terms of signless Laplacian spectral radius with large minimum degree.
Given a simple graph $G$, denote by $Delta(G)$, $delta(G)$, and $chi(G)$ the maximum degree, the minimum degree, and the chromatic index of $G$, respectively. We say $G$ is emph{$Delta$-critical} if $chi(G)=Delta(G)+1$ and $chi(H)le Delta(G)$ for eve
The degree-based entropy of a graph is defined as the Shannon entropy based on the information functional that associates the vertices of the graph with the corresponding degrees. In this paper, we study extremal problems of finding the graphs attain
Let $vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $vec{T}_4$s, and this bound is asymptotically tight. W