arXiv subjects
Publications and source records attributed to Shenggui Zhang.
A weighted graph is a graph in which every edge is assigned a non-negative real number. In a weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is the sum of the weights of the edges incident with it. In this paper we give three weighted degree conditions for the existence of heavy or Hamilton paths with one or two given end-vertices in 2-connected weighted graphs.
Let $G$ be a $k$-connected graph with $k\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree of the vertices other than $x$ and $z$. Further, with this result, we prove that: If $G$ has $n$ vertices and $m$ edges, then it contains a cycle of length at least $2m/(n-1)$ passing through its any $k-1$ specified vertices. Our results generalize a theorem of Fan on the existence of long paths and a classical theorem of Erdös and Gallai on the existence of long cycles under the average degree condition.
In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.
Computer or communication networks are so designed that they do not easily get disrupted under external attack and, moreover, these are easily reconstructible if they do get disrupted. These desirable properties of networks can be measured by various graph parameters, such as connectivity, toughness, scattering number, integrity, tenacity, rupture degree and edge-analogues of some of them. Among these parameters, the tenacity and rupture degree are two better ones to measure the stability of a network. In this paper we consider two extremal problems on the tenacity of graphs: Determine the minimum and maximum tenacity of graphs with given order and size. We give a complete solution to the first problem, while for the second one, it turns out that the problem is much more complicated than that of the minimum case. We determine the maximum tenacity of trees and unicyclic graphs with given order and show the corresponding extremal graphs. These results are helpful in constructing stable networks with lower costs. The paper concludes with a discussion of a related problem on the edge vulnerability parameters of graphs.
Let $G$ be a graph on $n$ vertices. A vertex of $G$ with degree at least $n/2$ is called a heavy vertex, and a cycle of $G$ which contains all the heavy vertices of $G$ is called a heavy cycle. In this paper, we characterize the graphs which contain no heavy cycles. For a given graph $H$, we say that $G$ is $H$-\emph{heavy} if every induced subgraph of $G$ isomorphic to $H$ contains two nonadjacent vertices with degree sum at least $n$. We find all the connected graphs $S$ such that a 2-connected graph $G$ being $S$-heavy implies any longest cycle of $G$ is a heavy cycle.
Let $G$ be a graph on $n$ vertices. An induced subgraph $H$ of $G$ is called heavy if there exist two nonadjacent vertices in $H$ with degree sum at least $n$ in $G$. We say that $G$ is $H$-heavy if every induced subgraph of $G$ isomorphic to $H$ is heavy. For a family $\mathcal{H}$ of graphs, $G$ is called $\mathcal{H}$-heavy if $G$ is $H$-heavy for every $H\in\mathcal{H}$. In this paper we characterize all connected graphs $R$ and $S$ other than $P_3$ (the path on three vertices) such that every 2-connected $\{R,S\}$-heavy graph is Hamiltonian. This extends several previous results on forbidden subgraph conditions for Hamiltonian graphs.