Graph Of Order Research Articles

The maximum number of cliques in graphs with prescribed order, circumference and minimum degree

Erdős determined the maximum size of a nonhamiltonian graph of order n and minimum degree at least k in 1962. Recently, Ning and Peng generalized Erdős’s work and gave the maximum number of s-cliques hs(n,c,k) of graphs with prescribed order n, circumference c and minimum degree at least k. But for some quadruples n,c,k,s the maximum number of s-cliques is not attained by a graph of minimum degree k. For example, h2(15,14,3)=77 is attained by a unique graph of minimum degree 7, not 3. In this paper we obtain more precise information by determining the maximum number of s-cliques of a graph with prescribed order, circumference and minimum degree. As a corollary of our main result, we determine the maximum size of a k-connected graph with given order and circumference. Consequently we solve the corresponding problem for longest paths.

Semisymmetric graphs of order twice prime powers with the same prime valency

Semisymmetric graphs of order twice prime powers with the same prime valency

On regular d-handicap tournaments

A k-regular d-handicap tournament is an incomplete tournament in which n teams, ranked according to the natural numbers, play exactly k < n − 1 different teams exactly once and the strength of schedule of the i t h ranked team is d more than the ( i − 1) s t ranked team for some d ≥ 1 . That is, strength of schedules increase arithmetically by d with strength of team. A d-handicap distance antimagic labeling of a graph G = ( V , E ) of order n is a bijection ℓ : V → {1,2,…, n } with induced weight function w ( x i )= Σ x j ∈ N ( x i ) l ( x j ) such that ℓ( x i )= i and the sequence of weights w ( x 1 ), w ( x 2 ),…, w ( x n ) forms an arithmetic sequence with difference d ≥ 1. A graph G which admits such a labeling is called a d-handicap graph . Constructing a k -regular d -handicap tournament on n teams is equivalent to finding a k -regular d -handicap graph of order n . For d = 1 and n even, the existence has recently been completely settled for all pairs ( n , k ) , and some results are known for d = 2 . For d > 2, the only known result is restricted to the case where n is divisible by 2 d + 2 . In this paper, we construct infinite families of d -handicap graphs where the order is not restricted to a power of 2.

Zonal graphs of small cycle rank

A zonal labeling of a plane graph G is an assignment of the two nonzero elements of the ring Z 3 of integers modulo 3 to the vertices of G such that the sum of the labels of the vertices on the boundary of each region of G is the zero element of Z 3 . A plane graph possessing such a labeling is a zonal graph. There is a connection between zonal labelings of connected bridgeless cubic plane graphs and the Four Color Theorem. Zonal labelings of cycles play a role in this connection. The cycle rank of a connected graph of order n and size m is m − n + 1 . Thus, cycles have cycle rank 1. All zonal connected graphs of cycle rank at most 2 are determined.

Disjunctive domination in graphs with minimum degree at least two

A set D of vertices in G is a disjunctive dominating set in G if every vertex not in D is adjacent to a vertex of D or has at least two vertices in D at distance 2 from it in G. The disjunctive domination number, γ2d(G), of G is the minimum cardinality of a disjunctive dominating set in G. In this paper, we show that if G be a graph of order at least 3, δ(G)≥2 and with no component isomorphic to any of eight forbidden graphs, then γ2d(G)≤|G|3. Moreover, we provide an infinite family of graphs attaining this bound. In addition, we also study the case that G is a claw-free graph with minimum degree at least two.

Two-arc-transitive graphs of odd order: I

Two-arc-transitive graphs of odd order: I

Turán number for odd‐ballooning of trees

AbstractThe Turán number is the maximum number of edges in an ‐free graph on vertices. Let be any tree. The odd‐ballooning of , denoted by , is a graph obtained by replacing each edge of with an odd cycle containing the edge, and all new vertices of the odd cycles are distinct. In this paper, we determine the exact value of for sufficiently large and being good, which generalizes all the known results on for being a star, due to Erdős, Füredi, Gould, and Gunderson (1995), Hou, Qiu, and Liu (2018), and Yuan (2018), and provides some counterexamples with chromatic number three to a conjecture of Keevash and Sudakov (2004), on the maximum number of edges not in any monochromatic copy of in a 2‐edge‐coloring of a complete graph of order .

The forcing geodetic global domination number of a graph

Let [Formula: see text] be a connected graph and [Formula: see text] be a minimum geodetic global dominating set of [Formula: see text]. A subset [Formula: see text] is called a forcing subset for [Formula: see text] if [Formula: see text] is the unique minimum geodetic global dominating set containing [Formula: see text]. A forcing subset for [Formula: see text] of minimum cardinality is a minimum forcing subset of [Formula: see text]. The forcing geodetic global domination number of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum forcing subset of S. The forcing geodetic global domination number of [Formula: see text], denoted by [Formula: see text], is [Formula: see text], where the minimum is taken over all minimum geodetic global dominating sets [Formula: see text] in [Formula: see text]. The forcing geodetic global domination number of some standard graphs are determined. Some of its general properties are studied. It is shown that for every pair of positive integers a and b with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text]. The geodetic global domination number of join of graphs is also studied. Connected graphs of order [Formula: see text] with geodetic global domination number 2 are characterized. It is proved that, for a connected graph [Formula: see text] with [Formula: see text]. Then [Formula: see text] and characterized connected graphs for which the lower and the upper bounds are sharp.

A Note on Isolate Domination Number of a Cubic Graph

In this note we provide a solution to the problem “Find a structural characterization of cubic graph for which the isolate domination number equals one plus its domination number.” We show that if G is a cubic graph of order n and if 6 | n, then the isolate domination number of G is the same as the domination number of G. We also prove that if G is a connected cubic graph with diam(G) > 2, then the isolate domination number is the same as the domination number.

On Aα spectral extrema of graphs forbidding even cycles

Given a graph G, for a real number α∈[0,1], Nikiforov (2017) proposed the Aα-matrix of G as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and the degree diagonal matrix of G, respectively. The largest eigenvalue of Aα(G) is called the Aα-index of G. For n>k, let Sn,k be the join of a clique on k vertices with an independent set of n−k vertices. Then Sn,k+ denotes the graph obtained from Sn,k by adding an edge to connect two vertices of degree k. Very recently, Cioabǎ, Desai and Tait resolved the Nikiforov's conjecture: For fixed k≥2, and sufficiently large n, the {C2k+1,C2k+2}-free graph of order n with maximum adjacency spectral radius is Sn,k and the C2k+2-free graph of order n with maximum adjacency spectral radius is Sn,k+. In this paper, we confirm the Aα-spectral version of this conjecture: For fixed k≥2,0<α<1 and n≥324k6(k+1)2α6(1−α)2, the {C2k+1,C2k+2}-free graph of order n with maximum Aα-index is Sn,k and the C2k+2-free graph of order n with maximum Aα-index is Sn,k+.

A note on the bounds for the spectral radius of graphs

Let G=(V,E) be a finite undirected graph of order n and of size m. Let Δ and δ be the largest and the smallest degree of G, respectively. The spectral radius of G is the largest eigenvalue of the adjacency matrix of the graph G. In this note we give new bounds on the spectral radius of {C3,C4}-free graphs in terms of m,n,Δ and δ. Computer search shows that in most of the cases the bounds derived in this note are better than the existing bounds.

Spectral radius and spanning trees of graphs

For any integer k≥2, a spanning k-ended-tree is a spanning tree with at most k leaves. Motivated by the degree sum condition of Broersma and Tuinstra (1998) [4], we provide tight spectral conditions to guarantee the existence of a spanning k-ended-tree in a connected graph of order n with extremal graphs being characterized. Let T be a spanning tree of a connected graph. The leaf degree of T is the maximum number of leaves adjacent to v in T for any v∈V(T). By adopting Kaneko's theorem (2001) [17], we also present tight spectral conditions for the existence of a spanning tree with leaf degree at most k in a connected graph and determine extremal graphs, where k≥1 is an integer.

On Graphs of Order n with Metric Dimension $$n-4$$

On Graphs of Order n with Metric Dimension $$n-4$$

Maximizing the Mostar index for bipartite graphs and split graphs

Došlić et al. defined the Mostar index of a graph G as ∑uv∈E(G)|nG(u,v)−nG(v,u)|, where, for an edge uv of G, the term nG(u,v) denotes the number of vertices of G that have a smaller distance in G to u than to v. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order n is at most 318n3, and that the Mostar index of split graphs of order n is at most 427n3.

Stability of Woodall's Theorem and Spectral Conditions for Large Cycles

In the 1970s, Erdős asked how many edges are needed in a graph on $n$ vertices, to ensure the existence of a cycle of length exactly $n-k$. In this paper, we consider the spectral analog of Erdős' problem. Indeed, the problem of determining tight spectral radius conditions for cycles of length $\ell$ in a graph of order $n$ for each $\ell \in[3,n]$ seems very difficult. We determine tight spectral radius conditions for $C_{\ell}$ where $\ell$ belongs to an interval of the form $[n-\Theta(\sqrt{n}),n]$. As a main tool, we prove a stability result of a theorem due to Woodall, which states that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is an integer, if $e(G)&gt;\binom{n-k-1}{2}+\binom{k+2}{2}$ then $G$ contains a $C_{\ell}$ for each $\ell\in [3,n-k]$. We prove a tight spectral condition for the circumference of a $2$-connected graph with a given minimum degree, of which the main tool is a stability version of a 1976 conjecture of Woodall on circumference of a $2$-connected graph with a given minimum degree proved by Ma and the second author. We also give a brief survey on this area and point out where we are and our predicament.

Unbalanced Spanning Subgraphs in Edge Labeled Complete Graphs

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $\Delta$ and with $m(G)$ edges. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number of edges with label $+1$ in $G'$ is at least $\left(d+\frac{\min\left\{ 2-d-2\sqrt{1-d},\sqrt{d}-d\right\}}{2\Delta+1}-O\left(\frac{1}{n}\right)\right)m(G)$, that is, this number visibly exceeds its expected value $d\cdot m(G)$ when considering a uniformly random copy of $G$ in $K$. For $d=\frac{1}{2}$, and $\Delta\leq 2$, we present more detailed results.

A spectral condition for the existence of cycles with consecutive odd lengths in non-bipartite graphs

A graph G is called H-free, if it does not contain H as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Turán type problem: what is the maximum spectral radius of an H-free graph of order n? In this paper, we consider the Brualdi-Solheid-Turán type problem for non-bipartite graphs. Let Ka,b•K3 denote the graph obtained by identifying a vertex of Ka,b in the part of size b and a vertex of K3. We prove that if G is a non-bipartite graph of order n satisfying ρ(G)≥ρ(K⌈n−22⌉,⌊n−22⌋•K3), then G contains all odd cycles C2l+1 for each integer l∈[2,k] unless G≅K⌈n−22⌉,⌊n−22⌋•K3, provided that n is sufficiently large with respect to k. This resolves the problem posed by Guo, Lin and Zhao (2021).

The polynomial reconstruction problem: The first 50 years

The problem of reconstructing the characteristic polynomial of a graph of order at least 3 from the collection of characteristic polynomials of its vertex-deleted subgraphs was posed by Cvetković in 1973 as a spectral counter part to the well-known Ulam's reconstruction conjecture. Over the last 50 years, this problem has received notable attention, many positive results have been obtained, but in the general case the problem is still unresolved. In particular, no counter example is found in literature. In this expository paper we survey classical and some more recent results concerning the polynomial reconstruction problem, discuss some related problems, variations and generalizations.

Clique Centrality and Global Clique Centrality of Graphs

We formally introduce in this paper two parameters in graph theory, namely, clique centrality and global clique centrality. Let G be a finite, simple and undirected graph of order n. A clique in G is a nonempty subset W \(\subseteq\) V (G) such that the subgraph \(\langle\)W\(\rangle\)G induced by W is complete. The maximum size of any clique containing vertex u \(\in\) V (G) is called the clique centrality of u in G. Normalizing the sum of the clique centralities of all the vertices of G will lead us to the global clique centrality of G, whose value ranges from \(\frac{1}{m}\) to 1. In this paper, we study some general properties of the global clique centrality and then evaluate it for some parameterized families of graphs.

Diameter, edge-connectivity, and C4-freeness

Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that d≤(2n−3)/5 for every connected C4-free graph of order n, diameter d, and edge-connectivity at least 3, which is best possible up to a small additive constant. For edge-connectivity at least 4, we improve this to d≤(n−1)/3. Furthermore, adapting a construction due to Erdős, Pach, Pollack, and Tuza, for an odd prime power q at least 7, and every positive integer k, we show the existence of a connected C4-free graph of order n=(q2+q−1)k+1, diameter d=4k, and edge-connectivity λ at least q−6, that is, d≥4(n−1)/(λ2+O(λ)) for the constructed graph.