Connected N-vertex Graph Research Articles

On spectral extrema of graphs with given order and dissociation number

Identifying the graph with maximum or minimum spectral radius among a given class of graphs is a central problem in extremal spectral graph theory, known as the Brualdi–Solheid problem. For a graph G=(VG,EG), a subset S⊆VG is called a maximum dissociation set if the induced subgraph G[S] does not contain P3 as its subgraph, and the subset has maximum cardinality. The dissociation number of G is the number of vertices in a maximum dissociation set of G. In this paper, we first characterize all the connected graphs (resp. bipartite graphs, trees) having maximum spectral radius among connected graphs (resp. bipartite graphs, trees) with given order and dissociation number. Secondly, we show that the connected n-vertex graph with dissociation number φ having minimum spectral radius is a tree, where φ≥23n. Finally, we determine the graphs having minimum spectral radius with fixed order n and dissociation number φ∈{2,⌈2n/3⌉,n−1,n−2}.

Independent domination of graphs with bounded maximum degree

An independent dominating set of a graph, also known as a maximal independent set, is a set S of pairwise non-adjacent vertices such that every vertex not in S is adjacent to some vertex in S. We prove that for Δ=4 or Δ≥6, every connected n-vertex graph of maximum degree at most Δ has an independent dominating set of size at most (1−Δ⌊Δ2/4⌋+Δ)(n−1)+1. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most (1−Δ⌊Δ2/4⌋+Δ)n.

Metric dimension and pattern avoidance in graphs

In this paper, we prove a number of results about pattern avoidance in graphs with bounded metric dimension or edge metric dimension. We show that the maximum possible number of edges in a graph of diameter D and edge metric dimension k is at most (⌊2D3⌋+1)k+k∑i=1⌈D3⌉(2i)k−1, sharpening the bound of k2+kDk−1+Dk from Zubrilina (2018). We also show that the maximum value of n for which some graph of metric dimension ≤k contains the complete graph Kn as a subgraph is n=2k. We prove that the maximum value of n for which some graph of metric dimension ≤k contains the complete bipartite graph Kn,n as a subgraph is 2Θ(k). Furthermore, we show that the maximum value of n for which some graph of edge metric dimension ≤k contains K1,n as a subgraph is n=2k. We also show that the maximum value of n for which some graph of metric dimension ≤k contains K1,n as a subgraph is 3k−O(k).In addition, we prove that the d-dimensional grids ∏i=1dPri have edge metric dimension at most d. This generalizes two results of Kelenc et al. (2016), that non-path grids have edge metric dimension 2 and that d-dimensional hypercubes have edge metric dimension at most d. We also provide a characterization of n-vertex graphs with edge metric dimension n−2, answering a question of Zubrilina. As a result of this characterization, we prove that any connected n-vertex graph G such that edim(G)=n−2 has diameter at most 5. More generally, we prove that any connected n-vertex graph with edge metric dimension n−k has diameter at most 3k−1.

Isolation of Cycles

For any graph G, let $$\iota _{\mathrm{c}}(G)$$ denote the size of a smallest set D of vertices of G such that the graph obtained from G by deleting the closed neighbourhood of D contains no cycle. We prove that if G is a connected n-vertex graph that is not a triangle, then $$\iota _{\mathrm{c}}(G) \le n/4$$. We also show that the bound is sharp. Consequently, this settles a problem of Caro and Hansberg.

Fixed-Parameter Tractable Algorithm and Polynomial Kernel for Max-Cut Above Spanning Tree

Every connected graph on n vertices has a cut of size at least n − 1. We call this bound the spanning tree bound. In the Max-Cut Above Spanning Tree (Max-Cut-AST) problem, we are given a connected n-vertex graph G and a non-negative integer k, and the task is to decide whether G has a cut of size at least n − 1 + k. We show that Max-Cut-AST admits an algorithm that runs in time $\mathcal {O}(8^{k}n^{\mathcal {O}(1)})$, and hence it is fixed parameter tractable with respect to k. Furthermore, we show that Max-Cut-AST has a polynomial kernel of size $\mathcal {O}(k^{5})$.

Extremal problems on weak Roman domination number

We show that the weak Roman domination number of a connected n-vertex graph is at most 2n3 and characterize the graphs achieving equality. In addition, we provide a constructive characterization of the trees for which the weak Roman domination number equals the Roman domination number and reveal several structural properties of these trees. This answers a problem posed in M. Chellali et al. (2014) [4].

Solution to a conjecture on a Nordhaus–Gaddum type result for the Kirchhoff index

Let G be a connected graph. The resistance distance between any two vertices of G is defined as the net effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index of G, denoted by Kf(G), is the sum of resistance distances between all pairs of vertices in G. In [28], it was conjectured that for a connected n-vertex graph G with a connected complement G¯,Kf(G)+Kf(G¯)≤n3−n6+n∑k=1n−11n−4sin2kπ2n,with equality if and only if G or G¯ is the path graph Pn. In this paper, by employing combinatorial and electrical techniques, we show that the conjecture is true except for a complementary pair of small graphs on five vertices.

Dendrimer Graphs as Thorn Graphs and Their Topological Edge Properties

Let G be a connected n-vertex graph with the vertex set V(G) = {v 1, v 2, …, v n } and let P = (p 1, p 2, …, p n ) be an n-tuple of nonnegative integers. The thorn graph G P is the graph obtained by attaching p i new vertices of degree one to the vertex v i of G, for i = 1, 2, …, n. In this paper, relations between the edge-Wiener indices of G and G P have been established and several special cases of these results have been examined. Results are applied to obtain closed formulas for the terminal Wiener index, the first and the second edge-Wiener indices of an infinite family of dendrimers by considering them as thorn graphs of simpler dendrimers.

Adding Edges to Increase the Chromatic Number of a Graph

If n ⩾ k + 1 and G is a connected n-vertex graph, then one can add $\binom{k}{2}$ edges to G so that the resulting graph contains the complete graph Kk+1. This yields that for any connected graph G with at least k + 1 vertices, one can add $\binom{k}{2}$ edges to G so that the resulting graph has chromatic number > k. A long time ago, Bollobás suggested that for every k ⩾ 3 there exists a k-chromatic graph Gk such that after adding to it any $\binom{k}{2}$ − 1 edges, the chromatic number of the resulting graph is still k. In this note we prove this conjecture.

Cop vs. Gambler

We consider a variation of cop vs. robber on graph in which the robber is not restricted by the graph edges; instead, he picks a time-independent probability distribution on V(G) and moves according to this fixed distribution. The cop moves from vertex to adjacent vertex with the goal of minimizing expected capture time. Players move simultaneously. We show that when the gambler’s distribution is known, the expected capture time (with best play) on any connected n-vertex graph is exactly n. We also give bounds on the (generally greater) expected capture time when the gambler’s distribution is unknown to the cop.

A characterization of graphs with rank 5

The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: what is the structure of a connected graph G with rank 5? or equivalently, what is the structure of a connected n-vertex graph G whose adjacency matrix has nullity n-5? In this paper, we completely characterize connected graphs G whose adjacency matrix has rank 5.

Extremal Problems for Roman Domination

A Roman dominating function of a graph G is a labeling $f\colon\,V(G)\to\{0,1,2\}$ such that every vertex with label 0 has a neighbor with label 2. The Roman domination number $\gamma_R(G)$ of G is the minimum of $\sum_{v\in V(G)}f(v)$ over such functions. Let G be a connected n-vertex graph. We prove that $\gamma_R(G)\leq4n/5$, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for $\gamma_R(G)+\gamma_R(\overline{G})$ and $\gamma_R(G)\gamma_R(\overline{G})$, improving known results for domination number. We prove that $\gamma_R(G)\leq8n/11$ when $\delta(G)\geq2$ and $n\geq9$, and this is sharp.

Spanning Trees with Many Leaves

A connected graph having large minimum vertex degree must have a spanning tree with many leaves. In particular, let $l( n,k )$ be the maximum integer m such that every connected n-vertex graph with minimum degree at least k has a spanning tree with at least m leaves. Then $l( n,3 )\geqq n/4 + 2, l( n,4 )\geqq ( 2n + 8 )/5,$ and $l( n,k )\leqq n - 3\lfloor n/( k + 1 ) \rfloor + 2$ for all k. The lower bounds are proved by an algorithm that constructs a spanning tree with at least the desired number of leaves. Finally, $l( n,k )\geqq ( 1 - b \ln k/k )n$ for large k, again proved algorithmically, where b is any constant exceeding 2.5.