On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

Some graft transformations and its applications on the distance spectral radius of a graph

Let D(G)=(d_i,j)_n×n denote the distance matrix of a connected graph G with order n, where d_ij is equal to the distance between v_i and v_j in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ?(G). In this paper, some graft transformations that decrease or increase ?(G) are given. With them, for the graphs with both order n and k pendant vertices, the extremal graphs with the minimum distance spectral radius are completely characterized; the extremal graph with the maximum distance spectral radius is shown to be a dumbbell graph (obtained by attaching some pendant edges to each pendant vertex of a path respectively) when 2≤k≤n−2; for k=1,2,3,n−1, the extremal graphs with the maximum distance spectral radius are completely characterized.     

Graphs satisfying inequality θ(G)θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G²)θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T²) for a tree T.     

Diameters of iterated clique graphs of chordal graphs

The clique graph K(G) of a graph is the intersection graph of maximal cliques of G. The iterated clique graph Kⁿ(G) is inductively defined as K(K^n?1(G)) and K¹(G) = K(G). Let the diameter diam(G) be the greatest distance between all pairs of vertices of G. We show that diam(Kⁿ(G)) = diam(G) — n if G is a connected chordal graph and n ≤ diam(G). This generalizes a similar result for time graphs by Bruce Hedman.     

Maximal and minimal entry in the principal eigenvector for the distance matrix of a graph

Let G=(V,E) be a simple, connected and undirected graph with vertex set V(G) and edge set E(G). Also let D(G) be the distance matrix of a graph G (Jane?i? et al., 2007) [13]. Here we obtain Nordhaus–Gaddum-type result for the spectral radius of distance matrix of a graph.A sharp upper bound on the maximal entry in the principal eigenvector of an adjacency matrix and signless Laplacian matrix of a simple, connected and undirected graph are investigated in Das (2009) [4] and Papendieck and Recht (2000) [15]. Generally, an upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries and without zero diagonal entries are investigated in Zhao and Hong (2002) [21] and Das (2009) [4], respectively. In this paper, we obtain an upper bound on minimal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. Moreover, we present the lower and upper bounds on maximal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs.     

An application of the Turán theorem to domination in graphs

A function f:V(G)→{+1,−1} defined on the vertices of a graph G is a signed dominating function if for any vertex v the sum of function values over its closed neighborhood is at least 1. The signed domination number γ_s(G) of G is the minimum weight of a signed dominating function on G. By simply changing “{+1,−1}” in the above definition to “{+1,0,−1}”, we can define the minus dominating function and the minus domination number of G. In this note, by applying the Turán theorem, we present sharp lower bounds on the signed domination number for a graph containing no (k+1)-cliques. As a result, we generalize a previous result due to Kang et al. on the minus domination number of k-partite graphs to graphs containing no (k+1)-cliques and characterize the extremal graphs.     

Some extremal results in cochromatic and dichromatic theory

For a graph G, the cochromatic number of G, denoted z(G), is the least m for which there is a partition of the vertex set of G having order m. where each part induces a complete or empty graph. We show that if {G_n} is a family of graphs where G_n has o(n² log²(n)) edges, then z(G_n) = o(n). We turn our attention to dichromatic numbers. Given a digraph D, the dichromatic number of D is the minimum number of parts the vertex set of D must be partitioned into so that each part induces an acyclic digraph. Given an (undirected) graph G, the dichromatic number of G, denoted d(G), is the maximum dichromatic number of all orientations of G. Let m be an integer; by d(m) we mean the minimum size of all graphs G where d(G) = m. We show that d(m) = θ(m² ln²(m)).     

An edge-grafting theorem on Laplacian spectra of graphs and its application

Let 𝒰(n,?d) be the class of unicyclic graphs on n vertices with diameter d. This article presents an edge-grafting theorem on Laplacian spectra of graphs. By applying this theorem, we determine the unique graph with the maximum Laplacian spectral radius in 𝒰(n,?d). This extremal graph is different from that for the corresponding problem on the adjacency spectral radius as done by Liu et al. [Q. Liu, M. Lu, and F. Tian, On the spectral radius of unicyclic graphs with fixed diameter, Linear Algebra Appl. 420 (2007), 449–457].     

Signed 2-independence in graphs

A function f : V→{−1,1} defined on the vertices of a graph G=(V,E) is a signed 2-independence function if the sum of its function values over any closed neighbourhood is at most one. That is, for every vV, f(N[v])1, where N[v] consists of v and every vertex adjacent to v. The weight of a signed 2-independence function is f(V)=∑f(v), over all vertices vV. The signed 2-independence number of a graph G, denoted α_s²(G), equals the maximum weight of a signed 2-independence function of G. In this paper, we establish upper bounds for α_s²(G) in terms of the order and size of the graph, and we characterize the graphs attaining these bounds. For a tree T, upper and lower bounds for α_s²(T) are established and the extremal graphs characterized. It is shown that α_s²(G) can be arbitrarily large negative even for a cubic graph G.     

On the eccentric distance sum of trees and unicyclic graphs

Let G be a simple connected graph with the vertex set V(G). The eccentric distance sum of G is defined as ξd(G)=_∑v∈V(G)ε(v)DG(v), where ε(v) is the eccentricity of the vertex v and DG(v)=_∑u∈V(G)d(u,v) is the sum of all distances from the vertex v. In this paper we characterize the extremal unicyclic graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum. In addition, we characterize the extremal trees with given diameter and minimal eccentric distance sum.     

Iterated clique graphs with increasing diameters

A simple argument by Hedman shows that the diameter of a clique graph G differs by at most one from that of K(G), its clique graph. Hedman described examples of a graph G such that diam(K(G)) = diam(G) + 1 and asked in general about the existence of graphs such that diam(Kⁱ(G)) = diam(G) + i. Examples satisfying this equality for i = 2 have been described by Peyrat, Rall, and Slater and independently by Balakrishnan and Paulraja. The authors of the former work also solved the case i = 3 and i = 4 and conjectured that such graphs exist for every positive integer i. The cases i ≥ 5 remained open. In the present article, we prove their conjecture. For each positive integer i, we describe a family of graphs G such that diam(Kⁱ(G)) = diam(G) + i. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 147–154, 1998     

Upper domination and upper irredundance perfect graphs

Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) =IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of a family of forbidden induced subgraphs, and show that the class of Γ-perfect graphs is a subclass of IR-perfect graphs and that the class of absorbantly perfect graphs is a subclass of Γ-perfect graphs. These results imply a number of known theorems on Γ-perfect graphs and IR-perfect graphs. Moreover, we prove a sufficient condition for a graph to be Γ-perfect and IR-perfect which improves a known analogous result.     

Rooted induced trees in triangle‐free graphs

For a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceilFor a graph G, let t(G) denote the maximum number of vertices in an induced subgraph of Gthat is a tree. Further, for a vertex v∈V(G), let t(G, v) denote the maximum number of vertices in an induced subgraph of Gthat is a tree, with the extra condition that the tree must contain v. The minimum of t(G) (t(G, v), respectively) over all connected triangle‐free graphs G(and vertices v∈V(G)) on nvertices is denoted by t₃(n) (t(n)). Clearly, t(G, v)?t(G) for all v∈V(G). In this note, we solve the extremal problem of maximizing |G| for given t(G, v), given that Gis connected and triangle‐free. We show that and determine the unique extremal graphs. Thus, we get as corollary that $t_3(n)\ge t_3^{\ast}(n) = \lceil {\frac{1}{2}}(1+{\sqrt{8n-7}})\rceil$, improving a recent result by Fox, Loh and Sudakov. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 206–209, 2010     

Clique-transversal sets in 4-regular claw-free graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by τ _c (G), is the minimum cardinality of a clique-transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.     

Iterating the branching operation on a directed graph

The branching operation D, defined by Propp, assigns to any directed graph G another directed graph D(G) whose vertices are the oriented rooted spanning trees of the original graph G. We characterize the directed graphs G for which the sequence δ(G) = (G, D(G), D²(G),…) converges, meaning that it is eventually constant. As a corollary of the proof we get the following conjecture of Propp: for strongly connected directed graphs G, δ(G) converges if and only if D²(G) = D(G). © 1997 John Wiley & Sons, Inc.     

Connectivity of the Mycielskian of a digraph

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), which is called the Mycielskian of G. This work investigates the vertex connectivity and arc connectivity of the Mycielskian of a digraph D. This generalizes the recent results due to Balakrishnan and Raj [R. Balakrishnan, S.F. Raj, Connectivity of the Mycielskian of a graph, Discrete Math, 308 (2008), 2607–2610].     

Semiharmonic trees and monocyclic graphs

A graph G is defined to be semiharmonic if there is a constant μ (necessarily a natural number) such that, for every vertex v, the number of walks of length 3 starting in v equals μd_G(v) where d_G(v) is the degree of v. We determine all finite semiharmonic trees and monocyclic graphs.     

Orientation distance graphs

For two nonisomorphic orientations D and D′ of a graph G, the orientation distance d_o(D,D′) between D and D′ is the minimum number of arcs of D whose directions must be reversed to produce an orientation isomorphic to D′. The orientation distance graph 𝒟_o(G) of G has the set 𝒪(G) of pairwise nonisomorphic orientations of G as its vertex set and two vertices D and D′ of 𝒟₀(G) are adjacent if and only if d_o(D,D′) = 1. For a nonempty subset S of 𝒪(G), the orientation distance graph 𝒟₀(S) of S is the induced subgraph 〈S〉 of 𝒟_o(G). A graph H is an orientation distance graph if there exists a graph G and a set S⊆ 𝒪(G) such that 𝒟_o(S) is isomorphic to H. In this case, H is said to be an orientation distance graph with respect to G. This paper deals primarily with orientation distance graphs with respect to paths. For every integer n ≥ 4, it is shown that 𝒟_o(P_n) is Hamiltonian if and only if n is even. Also, the orientation distance graph of a path of odd order is bipartite. Furthermore, every tree is an orientation distance graph with respect to some path, as is every cycle, and for n ≥ 3 the clique number of 𝒟_o(P_n) is 2 if n is odd and is 3 otherwise. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 230–241, 2001     

Minimum degree of minimal defect -extendable bipartite graphs

A near perfect matching is a matching saturating all but one vertex in a graph. If G is a connected graph and any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. If for any edge e in a defect n-extendable graph G, G−e is not defect n-extendable, then G is minimal defect n-extendable. The minimum degree and the connectivity of a graph G are denoted by δ(G) and κ(G) respectively. In this paper, we study the minimum degree of minimal defect n-extendable bipartite graphs. We prove that a minimal defect 1-extendable bipartite graph G has δ(G)=1. Consider a minimal defect n-extendable bipartite graph G with n≥2, we show that if κ(G)=1, then δ(G)≤n+1 and if κ(G)≥2, then 2≤δ(G)=κ(G)≤n+1. In addition, graphs are also constructed showing that, in all cases but one, there exist graphs with minimum degree that satisfies the established bounds.     

Decomposable graphs and definitions with no quantifier alternation

Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D₀(G) be the version of D(G) where we do not allow quantifier alternations in Φ. Define q₀(n) to be the minimum of D₀(G) over all graphs G of order n.We prove that for all n we have

log^*n−log^*log^*n−2≤q₀(n)≤log^*n+22,