Bipartite Graph Of Order Research Articles

On New Record Graphs Close to Bipartite Moore Graphs

The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and optimising one of the parameters given restrictions on some of the others. Here we focus on bipartite Moore graphs, that is, bipartite graphs attaining the optimum order, fixed either the degree/diameter or degree/girth. The fact that there are very few bipartite Moore graphs suggests the relaxation of some of the constraints implied by the bipartite Moore bound. First we deal with local bipartite Moore graphs. We find in some cases those local bipartite Moore graphs with local girths as close as possible to the local girths given by a bipartite Moore graph. Second, we construct a family of \((q+2)\)-bipartite graphs of order \(2(q^2+q+5)\) and diameter 3, for q a power of prime. These graphs attain the record value for \(q=9\) and improve the values for \(q=11\) and \(q=13\).

The spectral radius of graphs with given independence number

As a classic problem of spectral graph theory, Brualdi-Solheid problem asks which graph achieves the extremal (maximum or minimum) spectral radius for a given class of graphs. In this paper, we focus on this problem for graphs with fixed independence number. We first obtain the maximum spectral radius and unique extremal graph among all bipartite graphs of order n with independence number α . Secondly, we study the minimum spectral radius of graphs with fixed independence number. Let G n , α be the set of connected graphs of order n with independence number α . We show that tree has the minimum spectral radius over all graphs in G n , α provided that α ≥ ⌈ n 2 ⌉ , and determine all the extremal graphs in G n , n − 4 .

Diameter Three Orientability of Bipartite Graphs

In 2019, Czabarka, Dankelmann and Székely showed that for every undirected graph of order $n$, the minimum degree threshold for diameter two orientability is $\frac{n}{2}+ \Theta(\ln n)$. In this paper, we consider bipartite graphs and give a sufficient condition in terms of the minimum degree for such graphs to have oriented diameter three. We in particular prove that for balanced bipartite graphs of order $n$, the minimum degree threshold for diameter three orientability is $\frac{n}{4}+\Theta(\ln n)$.

On the critical difference of almost bipartite graphs

A set $$S\subseteq V$$ is independent in a graph $$G=\left( V,E\right) $$ if no two vertices from S are adjacent. The independence number $$\alpha (G)$$ is the cardinality of a maximum independent set, while $$\mu (G)$$ is the size of a maximum matching in G. If $$\alpha (G)+\mu (G)$$ equals the order of G, then G is called a König–Egerváry graph (Deming in Discrete Math 27:23–33, 1979; Sterboul in J Combin Theory Ser B 27:228–229, 1979). The number $$d\left( G\right) =\max \{\left| A\right| -\left| N\left( A\right) \right| :A\subseteq V\}$$ is called the critical difference of G (Zhang in SIAM J Discrete Math 3:431–438, 1990) (where $$N\left( A\right) =\left\{ v:v\in V,N\left( v\right) \cap A\ne \emptyset \right\} $$ ). It is known that $$\alpha (G)-\mu (G)\le d\left( G\right) $$ holds for every graph (Levit and Mandrescu in SIAM J Discrete Math 26:399–403, 2012; Lorentzen in Notes on covering of arcs by nodes in an undirected graph, Technical report ORC 66-16. University of California, Berkeley, CA, Operations Research Center, 1966; Schrijver in Combinatorial optimization. Springer, Berlin, 2003). In Levit and Mandrescu (Graphs Combin 28:243–250, 2012), it was shown that $$d(G)=\alpha (G)-\mu (G)$$ is true for every König–Egerváry graph. A graph G is (i) unicyclic if it has a unique cycle and (ii) almost bipartite if it has only one odd cycle. It was conjectured in Levit and Mandrescu (in: Abstracts of the SIAM conference on discrete mathematics, Halifax, Canada, p 40, abstract MS21, 2012, 3rd international conference on discrete mathematics, June 10–14, Karnatak University. Dharwad, India, 2013) and validated in Bhattacharya et al. (Discrete Math 341:1561–1572, 2018) that $$d(G)=\alpha (G)-\mu (G)$$ holds for every unicyclic non-König–Egerváry graph G. In this paper, we prove that if G is an almost bipartite graph of order $$n\left( G\right) $$ , then $$\alpha (G)+\mu (G)\in \left\{ n\left( G\right) -1,n\left( G\right) \right\} $$ . Moreover, for each of these two values, we characterize the corresponding graphs. Further, using these findings, we show that the critical difference of an almost bipartite graph G satisfies $$\begin{aligned} d(G)=\alpha (G)-\mu (G)=\left| \mathrm {core}(G)\right| -\left| N(\mathrm {core}(G))\right| , \end{aligned}$$ where by core $$\left( G\right) $$ we mean the intersection of all maximum independent sets.

Some generalizations of spectral conditions for 2s-hamiltonicity and 2s-traceability of bipartite graphs

ABSTRACT This paper mainly focuses on spectral conditions for the Hamiltonicity of balanced bipartite graphs and nearly balanced bipartite graphs. Let and be the spectral radius and signless Laplacian spectral radius of a graph G, respectively. One of our main results is the following theorem: Let G be a balanced bipartite graph of order and of minimum degree . If and , then G is -hamiltonian unless . If and , then G is -hamiltonian unless . This theorem generalizes the results in [B. L. Li, B. Ning, Spectral analogues of Erdos' and Moon–Moser's theorems on Hamilton cycles, Linear and Multilinear Algebra, 64 (2016) 2252–2269].

On the extremal graphs with respect to the total reciprocal edge-eccentricity

The total reciprocal edge-eccentricity of a graph G is defined as $$\xi ^{ee}(G)=\sum _{u\in V_G}\frac{d_G(u)}{\varepsilon _G(u)}$$, where $$d_G(u)$$ is the degree of u and $$\varepsilon _G(u)$$ is the eccentricity of u. In this paper, we first characterize the unique graph with the maximum total reciprocal edge-eccentricity among all graphs with a given number of cut vertices. Then we determine the k-connected bipartite graphs of order n with diameter d having the maximum total reciprocal edge-eccentricity. Finally, we identify the unique tree with the minimum total reciprocal edge-eccentricity among the n-vertex trees with given degree sequence.

Harary index of bipartite graphs

Let G be a connected graph with vertex set V ( G ) . The Harary index of a graph is defined as H ( G ) = ∑ u ≠ v 1/ d ( u , v ) , where d ( u , v ) denotes the distance between u and v . In this paper, we determine the extremal graphs with the maximum Harary index among all bipartite graphs of order n with a given matching number, with a given vertex-connectivity and with a given edge-connectivity, respectively.

A correction on the signed bad number

A signed bad function of [Formula: see text] is a function [Formula: see text] such that [Formula: see text] for every [Formula: see text] where [Formula: see text] is the closed neighborhood of [Formula: see text]. The signed bad number is [Formula: see text]. Ghameshlou et al. [A. N. Ghameshlou, A. Khodkar and S. M. Sheikholeslami, The signed bad numbers in graphs, Discrete Math. Algorithms Appl. 1 (2011) 33–41] proved that for any bipartite graph of order [Formula: see text], [Formula: see text]. But their proof has a gap and the bound is not correct in general. In this note, we modify their proof and show that for any bipartite graph of order [Formula: see text], [Formula: see text], and also we characterize the bipartite graphs attaining this bound.

Bounds for the $m$-Eternal Domination Number of a Graph

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard and that each vertex contains at most one guard). More than one guard is allowed to move in response to an attack. The $m$-eternaldomination number, $\edom(G)$, of a graph $G$ is the minimum number of guards needed to defend $G$ against any such sequence. We show that if $G$ is a connected graph with minimum degree at least~$2$ and of order~$n \ge 5$, then $\edom(G) \le \left\lfloor \frac{n-1}{2} \right\rfloor$, and this bound is tight. We also prove that if $G$ is a cubic bipartite graph of order~$n$, then $\edom(G) \le \frac{7n}{16}$.

On energy and Laplacian energy of bipartite graphs

Let G be a bipartite graph of order n with m edges. The energy E ( G ) of G is the sum of the absolute values of the eigenvalues of the adjacency matrix A. In 1974, one of the present authors established lower and upper bounds for E ( G ) in terms of n, m, and det A . Now, more than 40 years later, we correct some details of this result and determine the extremal graphs. In addition, an upper bound on the Laplacian energy of bipartite graphs in terms of n, m, and the first Zagreb index is obtained, and the extremal graphs characterized.

Decomposition of complete graphs into small generalized prisms

A prism is the Cartesian product Cm□P2. In other words, it is a graph consisting of two cycles of the same length whose corresponding vertices are joined by additional edges forming a matching. The problem of decomposition of complete graphs into prisms with 12 or 16 vertices was completely solved in [12]. In this paper we completely characterize the complete graphs that are decomposable into 3-regular bipartite graphs of order 12 or 16 that are a simple modification of prisms.

Signed Roman domination in graphs

In this paper we continue the study of Roman dominating functions in graphs. A signed Roman dominating function (SRDF) on a graph G=(V,E) is a function f:V?{?1,1,2} satisfying the conditions that (i) the sum of its function values over any closed neighborhood is at least one and (ii) for every vertex u for which f(u)=?1 is adjacent to at least one vertex v for which f(v)=2. The weight of a SRDF is the sum of its function values over all vertices. The signed Roman domination number of G is the minimum weight of a SRDF in G. We present various lower and upper bounds on the signed Roman domination number of a graph. Let G be a graph of order n and size m with no isolated vertex. We show that $\gamma _{\mathrm{sR}}(G) \ge\frac{3}{\sqrt{2}} \sqrt{n} - n$ and that ? sR(G)?(3n?4m)/2. In both cases, we characterize the graphs achieving equality in these bounds. If G is a bipartite graph of order n, then we show that $\gamma_{\mathrm{sR}}(G) \ge3\sqrt{n+1} - n - 3$ , and we characterize the extremal graphs.

Bounds for the Kirchhoff Index of Bipartite Graphs

A (m, n)‐bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell D(n, a, b) consists of the path Pn−a−b together with a independent vertices adjacent to one pendent vertex of Pn−a−b and b independent vertices adjacent to the other pendent vertex of Pn−a−b. In this paper, firstly, we show that, among (m, n)‐bipartite graphs (m ≤ n), the complete bipartite graph Km,n has minimal Kirchhoff index and the tree dumbbell D(m + n, ⌊n − (m + 1)/2⌋, ⌈n − (m + 1)/2⌉) has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order l, the complete bipartite graph K⌊l/2⌋,l−⌊l/2⌋ has minimal Kirchhoff index and the path Pl has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of (m, n)‐bipartite graphs and bipartite graphs of order l are obtained by computing the Kirchhoff index of these extremal graphs.

On permutation labeling

We determine all permutation graphs of order ⩽9. We prove that every bipartite graph of order ⩽50 is a permutation graph. We convert the conjecture stating that “every tree is a permutation graph” to be “every bipartite graph is a permutation graph”.

Sufficient conditions for bipartite graphs to be super- [formula omitted]-restricted edge connected

For a connected graph G = ( V , E ) , an edge set S ⊂ E is a k -restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k -restricted edge connectivity of G , denoted by λ k ( G ) , is defined as the cardinality of a minimum k -restricted edge cut. For U 1 , U 2 ⊂ V ( G ) , denote the set of edges of G with one end in U 1 and the other in U 2 by [ U 1 , U 2 ] . Define ξ k ( G ) = min { | [ U , V ( G ) ∖ U ] | : U ⊂ V ( G ) , | U | = k ≥ 1 and the subgraph induced by U is connected } . A graph G is λ k -optimal if λ k ( G ) = ξ k ( G ) . Furthermore, if every minimum k -restricted edge cut is a set of edges incident to a certain connected subgraph of order k , then G is said to be super- k -restricted edge connected or super- λ k for simplicity. Let k be a positive integer and let G be a bipartite graph of order n ≥ 4 with the bipartition ( X , Y ) . In this paper, we prove that: (a) If G has a matching that saturates every vertex in X or every vertex in Y and | N ( u ) ∩ N ( v ) | ≥ 2 for any u , v ∈ X and any u , v ∈ Y , then G is λ 2 -optimal; (b) If G has a matching that saturates every vertex in X or every vertex in Y and | N ( u ) ∩ N ( v ) | ≥ 3 for any u , v ∈ X and any u , v ∈ Y , then G is super- λ 2 ; (c) If the minimum degree δ ( G ) ≥ n + 2 k 4 , then G is λ k -optimal; (d) If the minimum degree δ ( G ) ≥ n + 2 k + 3 4 , then G is super- λ k .

The minimum number of vertices for a triangle-free graph with [formula omitted] is 11

It is well-known that the minimum number of vertices for a triangle-free 4-chromatic graph is 11, and the Grötzsch graph is just such a graph. In this paper, we show that every non-bipartite triangle-free graph G of order not greater than 10 has χ l ( G ) = 3 . Combined with a known result by Hanson et al. [D. Hanson, G. MacGillivray, B. Toft, Choosability of bipartite graphs, Ars Combin. 44 (1996) 183–192] that every bipartite graph of order not greater than 13 is 3-choosable, we conclude that the minimum number of vertices for a triangle-free graph with χ l ( G ) = 4 is also 11.

On the irregularity of bipartite graphs

The imbalance of an edge uv in a graph G is defined as |d(u)-d(v)|, where d(u) denotes the degree of u. The irregularity of G, denoted irr(G), is the sum of the edge imbalances taken over all edges in G. We determine the structure of bipartite graphs having maximum possible irregularity with given cardinalities of the partite sets and given number of edges. We then derive a corresponding result for bipartite graphs with given cardinalities of the partite sets and determine an upper bound on the irregularity of these graphs. In particular, we show that if G is a bipartite graph of order n with partite sets of equal cardinalities, then irr(G)⩽n3/27, while if G is a bipartite graph with partite sets of cardinalities n1 and n2, where n1⩾2n2, then irr(G)⩽irr(Kn1,n2).

Some results on placing bipartite graphs of maximum degree two

Abstract Bipartite graphs G = ( L , R ; E ) and H = ( L ′ , R ′ ; E ′ ) are bi-placeabe if there is a bijection f : L ∪ R → L ′ ∪ R ′ such that f ( L ) = L ′ and f ( u ) f ( v ) ∉ E ′ for every edge u v ∈ E ). We prove that if G and H are two bipartite graphs of order | G | = | H | = 2 p ⩾ 4 such that the sizes of G and H satisfy ‖ G ‖ ⩽ 2 p − 3 and ‖ H ‖ ⩽ 2 p − 2 , and the maximum degree of H is at most 2, then G and H are bi-placeable, unless G and H is one of easily recognizable couples of graphs. This result implies easily that for integers p and k 1 , k 2 , … , k l such that k i ⩾ 2 for i = 1 , … , l and k 1 + … + k 1 ⩽ p − 1 every bipartite balanced graph G of order 2p and size at least p 2 − 2 p + 3 contains mutually vertex disjoint cycles C 2 k 1 , … , C 2 k l , unless G = K 3 , 3 − 3 K 1 , 1 .

Maximal independent sets in bipartite graphs

AbstractA maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. In this paper, we determine the maximum number of maximal independent sets among all bipartite graphs of order n and the extremal graphs as well as the corresponding results for connected bipartite graphs. © 1993 John Wiley & Sons, Inc.