Graph Of Order Research Articles

Proof of a conjecture on dominating sets inducing large component in graphs with minimum degree two

The k-component domination number γk(G) of G is the minimum cardinality of a dominating set S of G such that each component of G[S] has order at least k. Clearly, γ1(G) is the domination number of G and γ2(G) is the total domination number of G. With a finite set of exceptional graphs, McCuaig and Shepherd (1989) [13] proved that γ1(G)≤25n and Henning (2000) [11] proved that γ2(G)≤47n respectively. As an extension, Alvarado et al. (2016) [1] posed a conjecture which says that if G is a connected graph of order n≥k+1 and δ(G)≥2, then γk(G)≤2kn2k+3 unless G belongs to a set of exceptional graphs. In this paper, we confirm the validity of the above conjecture. Moreover, we characterize those graphs of order n which are edge-minimal with respect to satisfying G connected, δ(G)≥2 and γk(G)≥2kn2k+3. This strengthens a result of Alvarado, Dantas, and Rautenbach in the same paper.

On hendecagonal circular ladder and its metric dimension

Let Γ = ( V , E ) be a connected graph of order n. Two vertices p and q in V are said to resolve by a vertex x ∈ V if d ( p , x ) ≠ d ( q , x ) . An ordered subset F = { k 1 , k 2 , k 3 , … , k l } of vertices in Γ is said to be resolving set if for every pair p, q of distinct vertices in V, we have ζ ( p | F ) ≠ ζ ( q | F ) , where ζ ( a | F ) = ( d ( a , k 1 ) , d ( a , k 2 ) , d ( a , k 3 ) , … , d ( a , k l ) ) is the l-code/metric coordinate representation of the vertex a with respect to the set F. The resolving set for Γ with minimum cardinality is known as metric basis for the graph Γ and the cardinality of metric basis is called as metric dimension of Γ. In this work, we demonstrate that for two families of convex polytopes which are closely linked, the metric dimension is constant.

A Logarithmic Bound for Simultaneous Embeddings of Planar Graphs

AbstractA set $${\mathcal {G}}$$ G of planar graphs on the same number n of vertices is called simultaneously embeddable if there exists a set P of n points in the plane such that every graph $$G \in {\mathcal {G}}$$ G ∈ G admits a (crossing-free) straight-line embedding with vertices placed at points of P. A conflict collection is a set of planar graphs of the same order with no simultaneous embedding. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether there exists a conflict collection of size 2. While this remains widely open, we give a short proof that for sufficiently large n there exists a conflict collection consisting of at most $$(3+o(1))\log _2(n)$$ ( 3 + o ( 1 ) ) log 2 ( n ) planar graphs on n vertices. This constitutes a double-exponential improvement over the previously best known bound of $$O(n\cdot 4^{n/11})$$ O ( n · 4 n / 11 ) for the same problem by Goenka et al. (Graphs Combin 39:100, 2023). Using our method we also provide a computer-free proof that for every integer $$n\in [107,193]$$ n ∈ [ 107 , 193 ] there exists a conflict collection of 30 planar n-vertex graphs, improving upon the previously smallest known conflict collection consisting of 49 graphs of order 11, which was found using heavy computer assistance. While the construction by Goenka et al. was explicit, our construction of a conflict collection of size $$O(\log n)$$ O ( log n ) is based on the probabilistic method and is thus only implicit. Motivated by this, for every large enough n we give a different, fully explicit construction of a collection of less than $$n^6$$ n 6 planar n-vertex graphs with no simultaneous embedding.

On 2-factors with long cycles in 3-connected claw-free graphs

For a graph H, let δ(H) be the minimum degree and let c(H) be the length of a longest cycle in H. A 2-factor of H is a spanning subgraph of H in which every component is a cycle. In [Discrete Math. 313 (2013) 1934-1943], Čada and Chiba asked if H is a 3-connected claw-free graph of order n with δ(H) sufficiently large, does H have a 2-factor F such that c(F)≥min⁡{9δ(H)−3,n}? In this paper, to answer their question, we show that if H is a 3-connected claw-free graph of order n with δ(H)≥15, then H has a 2-factor F such that c(F)≥min⁡{9δ(H)−3,n}.

Exploring Advanced Analysis Technique for Shallow Water Flow Models with Diverse Applications

The present article focuses on the analytical approach using fractional orders and its applicationin the dynamics of physical processes. Fractional order models align better with experimental data compared tonon-fractional ones. This study primarily focuses on employing the new approximate analytical method to solveshallow water models with fractional orders. Numerical examples within the Caputo fractional derivative showcasethe method’s application. Results for both integer and fractional orders are graphically depicted, demonstrating thefractional solutions’ closeness to actual data. Analysis of 3D and 2D fractional order graphs reveals convergencetoward integer order graphs as fractional derivatives approach non-fractional ones. This method shows promise fordirect application in solving targeted problems and can be easily adapted for other fractional nature problems.

Semisymmetric Graphs of Order 2 p3 with Valency p2

A simple undirected regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. For a semisymmetric graph [Formula: see text] of order [Formula: see text], [Formula: see text] a prime, it is well known that [Formula: see text] is bipartite with two biparts having equal size. The complete classification of such graphs has been given for the full automorphism group [Formula: see text] acting unfaithfully on at least one bipart of [Formula: see text], which shows that there is only one infinite family of such graphs with valency [Formula: see text]. The graphs of this kind have been determined when [Formula: see text] acts faithfully and primitively on at least one bipart of [Formula: see text], and thus there is only one remaining case for classifying such graphs of valency [Formula: see text], [Formula: see text] acting faithfully and imprimitively on both biparts of [Formula: see text], which is dealt with in this paper. As a result, there is only one infinite family of semisymmetric graphs of order [Formula: see text] with valency [Formula: see text].

Spectral Extremal Results on Trees

Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its independent set, where '$\nabla$' means the join product. For $n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioabă, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228-2239] showed that for $\ell\geq 6$ and sufficiently large $n$, if $\rho(G)\geq \rho(G_{n,\ell})$, then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree $F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with $|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and sufficiently large $n$. Consequently, $\rho(K_{q,n-q})\leq {\rm spex}(n,F)\leq \rho(G_{n,\ell})$, we further respectively characterize all trees $F$ with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of $G_{n,\ell}$.

Spectral extrema of 1-planar graphs

A graph G is called a 1-planar graph if G has a drawing in the plane, so that each of its edges is crossed at most once. In this paper, we prove that λ1(G)≤5+2n+5 for a 1-planar graph G of order n≥7. For n≥6.23×1018, we characterize the unique extremal graph with the maximum spectral radius among all 1-planar graphs of order n.

On the minimum size of graphs with given generalized connectivity

Let G be a connected graph and S⊆V(G) with |S|≥2. A tree T in G is called an S-tree if S⊆V(T). Two S-trees T1 and T2 are internally disjoint if E(T1)∩E(T2)=0̸ and V(T1)∩V(T2)=S. For an integer k≥2, the generalizedk-connectivity of a graph G, denoted by κk(G), is defined as κk(G)=min{κG(S):S⊆V(G) and |S|=k}, where κG(S) denotes the maximum number of pairwise internally disjoint S-trees in G. The generalized connectivity is a generalization of traditional connectivity. Let Gn be the class of connected graphs of order n and let f(n,k,t)=minG∈Gn{|E(G)|:κk(G)=t}. In this paper, we prove f(n,k,t)≥⌈t(t+2)2t+2n⌉ for 4≤k≤n and 1≤t≤n−⌈k2⌉. In particular, the lower bound is sharp when k=4 and t=2 (i.e., we explicitly provide a family of graphs fulfilling the bound in this case), improving the known results of Sun et al. (2021).

A short proof of Haemers' conjecture on the Seidel energy of graphs

Haemers' conjecture on the Seidel energy of graphs, states that the energy of the Seidel matrix of any graph of order n is at least 2n−2, where the equality holds for the complete graph. In this paper, we give a short simple proof of this conjecture.

Chen-Chvátal’s conjecture for graphs with restricted girth

A classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chvátal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3.

Local distance antimagic cromatic number of join product of graphs with cycles or paths

Let $G$ be a graph of order $p$ without isolated vertices. A bijection $f: V \to \{1,2,3,\dots,p\}$ is called a local distance antimagic labeling, if $w_f(u)\ne w_f(v)$ for every edge $uv$ of $G$, where $w_f(u)=\sum_{x\epsilon N(u)} {f(x)}$. The local distance antimagic chromatic number $\chi_{lda}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local distance antimagic labelings of $G$. In this paper, we determined the local distance antimagic chromatic number of some cycles, paths, disjoint union of 3-paths. We also determined the local distance antimagic chromatic number of join products of some graphs with cycles or paths.

Best possible upper bounds on the restrained domination number of cubic graphs

AbstractA dominating set in a graph is a set of vertices such that every vertex in is adjacent to a vertex in . A restrained dominating set of is a dominating set with the additional restraint that the graph obtained by removing all vertices in is isolate‐free. The domination number and the restrained domination number are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of . Let be a cubic graph of order . A classical result of Reed states that , and this bound is best possible. To determine the best possible upper bound on the restrained domination number of is more challenging, and we prove that .

All Ramsey critical graphs for large cycles vs a complete graph of order six

A new area of Graph theory emerged in the last few decades is the calculation of Star critical Ramsey numbers related to different classes of graphs. Formally, we will say that Kn → (G,H) if given any coloring of Kn there is a copy of G in the first color, red, or a copy of H in the second color, blue. The Ramsey number r(G,H) is defined as the smallest positive integer n such that Kn → (G,H). A closely related concept of Ramsey number is the Star critical Ramsey number r∗(G,H) defined as the largest value of k such that Kr(G,H)−1 ⊔ K1,k → (G,H). A two-coloring of Kr(G,H)−1 such that Kr(G,H)−1 ̸→ (G,H) is called a Ramsey critical coloring. A Ramsey critical r(G,H) graph is a graph induced by the first color of a Ramsey critical coloring. Lower bounds for Star critical Ramsey numbers are usually found with the aid on Ramsey critical graphs. The particular problem we handle in this paper, on Star critical Ramsey numbers, is based on a conjecture posed in 1973 by Bondy and Erdos relating to Ramsey numbers for large cycles versus complete graphs. Based on certain lemmas we present with proof, furthermore we show that there exist exactly sixty eight non-isomorphic Ramsey critical r(Cn,K6) graphs, when n ≥ 15

A bound on relative lengths of triangle-free graphs

For a 2-connected graph G, the relative length of G, denoted by diff(G), is the difference between the orders of a longest path and a longest cycle in G. This parameter is used as a measure to estimate how close a given graph is to a hamiltonian graph. Let σk(G) be the least value of the sums of degrees of vertices in independent sets of cardinality k. In 2008, Paulusma and Yoshimoto proved that a 2-connected triangle-free graph G of order n with σ4(G)≥n+2 satisfies diff(G)≤1 unless G is isomorphic to one exceptional graph G0. In this paper, we extend their result and prove that for an integer s with 23(n+4)<s≤n+2, a 2-connected triangle-free graph of order n with σ4(G)≥s satisfies diff(G)≤n+3−s unless s=σ4(G)=n+2 and G=G0.

2-coupon coloring of cubic graphs containing 3-cycle or 4-cycle

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex in S. Recently, the following question was proposed:“Is it true that every connected cubic graph containing a 3-cycle has two vertex disjoint total dominating sets?”In this paper, we give a negative answer to this question. Moreover, we prove that if we replace 3-cycle with 4-cycle the answer is affirmative. This implies every connected cubic graph containing a diamond (the complete graph of order 4 minus one edge) as a subgraph can be partitioned into two total dominating sets, a result that was proved in 2017.

Decycling cubic graphs

A set of vertices of a graph G is said to be decycling if its removal leaves an acyclic subgraph. The size of a smallest decycling set is the decycling number of G. Generally, at least ⌈(n+2)/4⌉ vertices have to be removed in order to decycle a cubic graph on n vertices. In 1979, Payan and Sakarovitch proved that the decycling number of a cyclically 4-edge-connected cubic graph of order n equals ⌈(n+2)/4⌉. In addition, they characterised the structure of minimum decycling sets and their complements. If n≡2(mod4), then G has a decycling set which is independent and its complement induces a tree. If n≡0(mod4), then one of two possibilities occurs: either G has an independent decycling set whose complement induces a forest of two trees, or the decycling set is near-independent (which means that it induces a single edge) and its complement induces a tree. In this paper we strengthen the result of Payan and Sakarovitch by proving that the latter possibility (a near-independent set and a tree) can always be guaranteed. Moreover, we relax the assumption of cyclic 4-edge-connectivity to a significantly weaker condition expressed through the canonical decomposition of 3-connected cubic graphs into cyclically 4-edge-connected ones. Our methods substantially use a surprising and seemingly distant relationship between the decycling number and the maximum genus of a cubic graph.

Laplacian coefficients, Kirchhoff index and the number of spanning trees of graphs

Let [Formula: see text], [Formula: see text], be a simple connected graph of order [Formula: see text] and size [Formula: see text]. Denote with [Formula: see text] eigenvalues of the Laplacian matrix [Formula: see text] of [Formula: see text]. The Kirchhoff index and the number of spanning trees of [Formula: see text] expressed in terms of Laplacian eigenvalues are given by [Formula: see text] and [Formula: see text], respectively. The characteristic polynomial of [Formula: see text] is given by [Formula: see text]. The first five Laplacian coefficients have been computed in the literature. In this study, we compute the sixth Laplacian coefficient of [Formula: see text]. Then, we use it to improve the previously obtained results on [Formula: see text] and [Formula: see text]. In addition, we present new Nordhaus–Gaddum-type inequalities for [Formula: see text] and [Formula: see text].

The Restricted Edge-Connectivity of Strong Product Graphs

The restricted edge-connectivity of a connected graph G, denoted by λ′(G), if it exists, is the minimum cardinality of a set of edges whose deletion makes G disconnected, and each component has at least two vertices. It was proved that λ′(G) exists if and only if G has at least four vertices and G is not a star. In this case, a graph G is called maximally restricted edge-connected if λ′(G)=ξ(G), and a graph G is called super restricted edge-connected if each minimum restricted edge-cut isolates an edge of G. The strong product of graphs G and H, denoted by G⊠H, is the graph with the vertex set V(G)×V(H) and the edge set {(x1,y1)(x2,y2)|x1=x2 and y1y2∈E(H); or y1=y2 and x1x2∈E(G); or x1x2∈E(G) and y1y2∈E(H)}. In this paper, we determine, for any nontrivial connected graph G, the restricted edge-connectivity of G⊠Pn, G⊠Cn and G⊠Kn, where Pn, Cn and Kn are the path, cycle and complete graph of order n, respectively. As corollaries, we give sufficient conditions for these strong product graphs G⊠Pn, G⊠Cn and G⊠Kn to be maximally restricted edge-connected and super restricted edge-connected.

Equi Neighbor Polynomial of Some Binary Graph Operations

Let \(G(V,E)\) be a simple graph of order \(n\) with vertex set \(V\) and edge set \(E\). Let \((u, v)\) denote an unordered vertex pair of distinct vertices of \(G\). For a vertex \(u \in G,\) let \(N(u)\) be the set of all vertices of \(G\) which are adjacent to \(u\) in \(G.\) Then for \(0\leq i \leq n-1\), the \(i\)-equi neighbor set of \(G\) is defined as: \(N_{e}(G,i)=\{(u,v):u, v\in V, u\neq v\) and \(|N(u)|=|N(v)|=i\}.\) The equi-neighbor polynomial \(N_{e}[G;x]\) of \(G\) is defined as \(N_{e}[G;x]=\sum_{i=0}^{(n-1)} |N_{e}(G,i)| x^{i}.\) In this paper we discuss the equi-neighbor polynomial of graphs obtained by some binary graph operations.