We prove a necessary and sufficient condition under which a connected graph has a connected P₃-path graph. Moreover, an analogous condition for connectivity of the Pₖ-path graph of a connected graph which does not contain a cycle of length smaller than k+1 is derived.

The eigenvalues of graphs are related to many of its combinatorial properties. In his fundamental work, Fiedler showed the close connections between the Laplacian eigenvalues and eigenvectors of a graph and its vertex-connectivity and edge-connectivity. We present some new results describing the connections between the spectrum of a regular graph and other combinatorial parameters such as its generalized connectivity, toughness, and the existence of spanning trees with bounded degree.

A tree is classified as being type I provided that there are two or more Perron branches at its characteristic vertex. The question arises as to how one might construct such a tree in which the Perron branches at the characteristic vertex are not isomorphic. Motivated by an example of Grone and Merris, we produce a large class of such trees, and show how to construct others from them. We also investigate some of the properties of a subclass of these trees. Throughout, we exploit connections between...

The $P_3$ intersection graph of a graph $G$ has for vertices all the induced paths of order 3 in $G$. Two vertices in $P_3\left(G\right)$ are adjacent if the corresponding paths in $G$ are not disjoint. A $w$-container between two different vertices $u$ and $v$ in a graph $G$ is a set of $w$ internally vertex disjoint paths between $u$ and $v$. The length of a container is the length of the longest path in it. The $w$-wide diameter of $G$ is the minimum number $l$ such that there is a $w$-container of length at most $l$ between any pair of different...

An edge $e$ of a $k$-connected graph $G$ is said to be $k$-contractible (or simply contractible) if the graph obtained from $G$ by contracting $e$ (i.e., deleting $e$ and identifying its ends, finally, replacing each of the resulting pairs of double edges by a single edge) is still $k$-connected. In 2002, Kawarabayashi proved that for any odd integer $k\ge 5$, if $G$ is a $k$-connected graph and $G$ contains no subgraph $D=K_1+(K_2\cup K_{1,2})$, then $G$ has a $k$-contractible edge. In this paper, by generalizing this result, we prove that for any integer...

The paper studies the domatic numbers and the total domatic numbers of graphs having cut-vertices.

If $x$ is a vertex of a digraph $D$, then we denote by $d^{+}\left(x\right)$ and $d^{-}\left(x\right)$ the outdegree and the indegree of $x$, respectively. A digraph $D$ is called regular, if there is a number $p\in \mathbb{N}$ such that $d^{+}\left(x\right)=d^{-}\left(x\right)=p$ for all vertices $x$ of $D$. A $c$-partite tournament is an orientation of a complete $c$-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether...

This paper is devoted to computational problems related to Markov chains (MC) on a finite state space. We present formulas and bounds for characteristics of MCs using directed forest expansions given by the Matrix Tree Theorem. These results are applied to analysis of direct methods for solving systems of linear equations, aggregation algorithms for nearly completely decomposable MCs and the Markov chain Monte Carlo procedures.

For an integer k ≥ 1, we say that a (finite simple undirected) graph G is k-distance-locally disconnected, or simply k-locally disconnected if, for any x ∈ V (G), the set of vertices at distance at least 1 and at most k from x induces in G a disconnected graph. In this paper we study the asymptotic behavior of the number of edges of a k-locally disconnected graph on n vertices. For general graphs, we show that this number is Θ(n2) for any fixed value of k and, in the special case of regular graphs,...

The strong product G₁ ⊠ G₂ of graphs G₁ and G₂ is the graph with V(G₁)×V(G₂) as the vertex set, and two distinct vertices (x₁,x₂) and (y₁,y₂) are adjacent whenever for each i ∈ 1,2 either $x_i=y_i$ or $x_i{y}_i\in E\left(G_i\right)$. In this note we show that for two connected graphs G₁ and G₂ the edge-connectivity λ (G₁ ⊠ G₂) equals minδ(G₁ ⊠ G₂), λ(G₁)(|V(G₂)| + 2|E(G₂)|), λ(G₂)(|V(G₁)| + 2|E(G₁)|). In addition, we fully describe the structure of possible minimum edge cut sets in strong products of graphs.