Complete Split Graphs Research Articles

Counting spanning trees of multiple complete split-like graph containing a given spanning forest

The multiple complete split-like graph MCSb,sa is the join of an empty graph K‾a and s copies of complete graph Kb. In this article, we obtain the formulas for the number of spanning trees of MCSb,sa containing a given spanning forest when s=1 and 2. Particularly, when s=1, our result derives the number of spanning trees of complete split graph containing a given spanning forest, thereby extending Moon's result [19].

The rotation distance of brooms

The associahedron A(G) of a graph G has the property that its vertices can be thought of as the search trees on G and its edges as the rotations between two search trees. If G is a simple path, then A(G) is the usual associahedron and the search trees on G are binary search trees. Computing distances in the graph of A(G), or equivalently, the rotation distance between two binary search trees, is a major open problem. Here, we consider the different case when G is a complete split graph. In that case, A(G) interpolates between the stellohedron and the permutohedron, and all the search trees on G are brooms. We show that the rotation distance between any two such brooms and therefore the distance between any two vertices in the graph of the associahedron of G can be computed in quasi-quadratic time in the number of vertices of G.

New Improved Bounds for Signless Laplacian Spectral Radius and Nordhaus-Gaddum Type Inequalities for Agave Class of Graphs

Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique Kc (the core) and η copies of Ks (the satellites) meeting in a common clique. They belong to the class of graphs of diameter two. Agave graphs Θ(2, 1, η) ∼= K2 ▽ (ηK1) belong to the general class of complete split graphs, where the graphs consist of a central clique K2 and η copies of K1 which are connected to all the nodes of the clique. They are the subclass of Core-satellite graphs. Let μ(G) be the spectral radius of the signless Laplacian matrix Q(G). In this paper, we have obtained the greatest lower bound and the least upper bound of signless Laplacian spectral radius of Agave graphs. These bounds have been expressed in terms of graph invariants like m the number of edges, n the number of vertices, δ the minimum degree, ∆ the maximum degree, and η copies of the satellite. We have made use of the approximation technique to derive these bounds. This unique approach can be utilized to determine the bounds for the signless Laplacian spectral radius of any general family of graphs. We have also obtained Nordhaus-Gaddum type inequality using the derived bounds.

Distance energy change of complete split graph due to edge deletion

The distance energy of a connected graph [Formula: see text] is the sum of absolute values of the eigenvalues of the distance matrix of [Formula: see text]. In this paper, we study how the distance energy of the complete split graph [Formula: see text] changes when an edge is deleted from it. The complete split graph [Formula: see text] consists of a clique on [Formula: see text] vertices and an independent set on [Formula: see text] vertices in which each vertex of the clique is adjacent to each vertex of the independent set. We prove that the distance energy of the complete split graph [Formula: see text] always increases when an edge is deleted from it.

On the distribution of eigenvalues of the reciprocal distance Laplacian matrix of graphs

The reciprocal distance Laplacian matrix of a connected graph G is defined as RDL(G) = RT(G) ? RD(G), where RT(G) is the diagonal matrix of reciprocal distance degrees and RD(G) is the Harary matrix. Since RDL(G) is a real symmetric matrix, we denote its eigenvalues as ?1(RDL(G)) ? ?2(RDL(G)) ?...? ?n(RDL(G)). The largest eigenvalue ?1(RDL(G)) of RDL(G) is called the reciprocal distance Laplacian spectral radius. In this article, we prove that the multiplicity of n as a reciprocal distance Laplacian eigenvalue of RDL(G) is exactly one less than the number of components in the complement graph G of G. We show that the class of the complete bipartite graphs maximize the reciprocal distance Laplacian spectral radius among all the bipartite graphs with n vertices. Also, we show that the star graph Sn is the unique graph having the maximum reciprocal distance Laplacian spectral radius in the class of trees with n vertices. We determine the reciprocal distance Laplacian spectrum of several well known graphs. We prove that the complete graph Kn, Kn ? e, the star Sn, the complete balanced bipartite graph Kn/2 ,n/2 and the complete split graph CS(n, ?) are all determined from the RDL-spectrum.

Reciprocal distance energy of complete multipartite graphs

In this paper, first we compute the energy of a special partitioned matrix under some cases. As a consequence, we obtain the reciprocal distance energy of the complete multipartite graph and also we give various other energies of complete multipartite graphs. Next, we show that among all complete [Formula: see text]-partite graphs on [Formula: see text] vertices, the complete split graph [Formula: see text] has minimum reciprocal distance energy and the reciprocal distance energy is maximum for the Turan graph [Formula: see text]. At last, it is shown that the reciprocal distance energy of the complete bipartite graph [Formula: see text] decreases under deletion of an edge if [Formula: see text], and increases if [Formula: see text]. Also, we show that the reciprocal distance energy of the complete tripartite graph does not increase under edge deletion.

Comparing eigenvector and degree dispersion with the principal ratio of a graph

The principal ratio of a graph is the ratio of the greatest and least entry of its principal eigenvector. Since the principal ratio compares the extreme values of the principal eigenvector it is sensitive to outliers. This can be problematic for graphs (networks) drawn from empirical data. To account for this we consider the dispersion of the principal eigenvector (and degree vector). More precisely, we consider the coefficient of variation of the aforementioned vectors, that is, the ratio of the vector's standard deviation and mean. We show how both of these statistics are bounded above by the same function of the principal ratio. Further, this bound is sharp for regular graphs. The goal of this paper is to show that the coefficient of variation of the principal eigenvector (and degree vector) can converge or diverge to the principal ratio in the limit. In doing so, we find an example of a graph family (the complete split graph) whose principal ratio converges to the golden ratio. We conclude with conjectures concerning extremal graphs of the aforementioned statistics.

Maxima of the Q-index: Forbidden a fan

The complete split graph, denoted by Sn,k, is the graph obtained by joining a copy of Kk to n−k isolated vertices. As a special threshold graph, it plays an important role in graph theory and spectral graph theory. Some known results tell us that if q(G)>q(Sn,k) then G can contain many kinds of subgraphs, such as paths, linear forests, friendship graphs, flowers, and so on (where q(G) is the Q-index of G). Motivated by these results, we are interested in finding larger subgraphs in G. In this paper, we show that if q(G)>q(Sn,k) then G contains a fan H2k, where H2k denotes the graph obtained by joining a vertex to a path of order 2k. This implies several previous results, and a conjecture proposed by Li and Peng.

Complexity of edge monitoring on some graph classes

In this paper, we study the complexity of the edge monitoring problem. A vertex v monitors an edge e if both endpoints together with v form a triangle in the graph. Given a graph G = ( V , E ) and a weight function on edges c where c ( e ) is the number of monitors that needs the edge e , the problem is to seek a minimum subset of monitors S such that every edge e in the graph is monitored by at least c ( e ) vertices in S . Therefore, we study the edge monitoring problem on several graph classes such as complete graphs, block graphs, cographs, split graphs, interval graphs and planar graphs. We also generalize the problem by adding weights on vertices.

ON THE PALETTE INDEX OF GRAPHS HAVING A SPANNING STAR

A proper edge coloring of a graph $G$ is a mapping $\alpha:E(G)\longrightarrow \mathbb{N}$ such that $\alpha(e)\not=\alpha(e')$ for every pair of adjacent edges $e$ and $e'$ in $G$. In a proper edge coloring of a graph $G$, the palette of a vertex $v \in V(G)$ is the set of colors assigned to the edges incident to $v$. The palette index of $G$ is the minimum number of distinct palettes occurring in $G$ among all proper edge colorings of $G$. A graph $G$ has a spanning star, if it has a spanning subgraph which is a star. In this paper, we consider the palette index of graphs having a spanning star. In particular, we give sharp upper and lower bounds on the palette index of these graphs. We also provide some upper and lower bounds on the palette index of the complete split and threshold graphs.

Anti-Ramsey numbers for cycles in the generalized Petersen graphs

For H⊆G, the anti-Ramsey number ar(G,H) is the maximum number of colors in an edge-coloring of G such that each subgraph isomorphic to H has at least two edges in the same color. The study of anti-Ramsey number ar(Kn,H) was introduced by Erdős et al. in 1973, and plentiful results were researched for some special graph H. Later, the problem was extended to ar(G,H) when replacing Kn by other graph G such as hypergraph, complete split graph, regular bipartite graph, triangulation and so on. In this paper, we consider a generalized Petersen graph Pn,k as the host graph and determine the exact anti-Ramsey numbers for cycles C5 and C6 in Pn,k, respectively.

Lucky \(k\)-polynomials of null and complete split graphs

The concept of Lucky colorings of a graph is used to introduce the notion of the Lucky \(k\)-polynomials of null graphs. We then give the Lucky \(k\)-polynomials for complete split graphs and generalized star graphs. Finally, further problems of research related to this concept are discussed.

On graphs with minimal distance signless Laplacian energy

Abstract For a simple connected graph G of order n having distance signless Laplacian eigenvalues ρ 1 Q ≥ ρ 2 Q ≥ ⋯ ≥ ρ n Q \rho _1^Q \ge \rho _2^Q \ge \cdots \ge \rho _n^Q , the distance signless Laplacian energy DSLE(G) is defined as D S L E ( G ) = ∑ i = 1 n | ρ i Q - 2 W ( G ) n | DSLE\left( G \right) = \sum\nolimits_{i = 1}^n {\left| {\rho _i^Q - {{2W\left( G \right)} \over n}} \right|} where W(G) is the Weiner index of G. We show that the complete split graph has the minimum distance signless Laplacian energy among all connected graphs with given independence number. Further, we prove that the graph Kk ∨ ( Kt∪ Kn−k−t), 1 ≤ t ≤ ⌊ n - k 2 ⌋ 1 \le t \le \left\lfloor {{{n - k} \over 2}} \right\rfloor has the minimum distance signless Laplacian energy among all connected graphs with vertex connectivity k.

Uniquely List Colorability of Complete Split Graphs

Uniquely List Colorability of Complete Split Graphs

Anti-Ramsey Number of Triangles in Complete Multipartite Graphs

An edge-colored graph is called rainbow if all its edges are colored distinct. The anti-Ramsey number of a graph family $${\mathcal {F}}$$ in the graph G, denoted by $$AR{(G,{\mathcal {F}})}$$ , is the maximum number of colors in an edge-coloring of G without rainbow subgraph in $${\mathcal {F}}$$ . The anti-Ramsey number for the short cycle $$C_3$$ has been determined in a few graphs. Its anti-Ramsey number in the complete graph can be easily obtained from the lexical edge-coloring. Gorgol considered the problem in complete split graphs which contains complete graphs as a subclass. In this paper, we study the problem in the complete multipartite graph which further enlarges the family of complete split graphs. The anti-Ramsey numbers for $$C_3$$ and $$C_3^{+}$$ in complete multipartite graphs are determined. These results contain the known results for $$C_3$$ and $$C_3^{+}$$ in complete and complete split graphs as corollaries.

A simple method for proving lower bounds in the zero-visibility cops and robber game

Cops and Robbers is a classical pursuit and evasion game in graph theory, which was introduced by Nowakowski and Winkler and independently by Quilliot. In this paper, we study the zero-visibility cops and robber game, which is a variant of Cops and Robbers. In the zero-visibility cops and robber game, the robber is invisible to the cops throughout the game. We introduce a simple method for proving lower bounds on the zero-visibility cop number. This lower bound method is based on a connection between the zero-visibility cop number and the matching number. Using this technique, we investigate graph joins, lexicographic products of graphs, complete multipartite graphs and split graphs. For each of these classes of graphs, we prove lower bounds and upper bounds on the zero-visibility cop number. We also present a linear time approximation algorithm for computing the lexicographic product of a tree and a graph G. The approximation ratio of this algorithm is bounded by $$|V(G)| / (\nu (G) + |V(G) {\setminus } V(\mathcal {M}(G))| )$$ , where V(G) is the vertex set of G, $$\nu (G)$$ is the matching number of G, $$\mathcal {M}(G)$$ is a maximum matching of G and $$V(\mathcal {M}(G))$$ is the vertex set of $$\mathcal {M}(G)$$ .

The signless Laplacian spectral radius of graphs with no intersecting triangles

Let Fk denote the k-fan consisting of k triangles which intersect in exactly one common vertex, and Sn,k the complete split graph of order n consisting of a clique on k vertices and an independent set on the remaining vertices in which each vertex of the clique is adjacent to each vertex of the independent set. In this paper, it is shown that Sn,k is the unique graph attaining the maximum signless Laplacian spectral radius among all graphs of order n containing no Fk, provided that k≥2 and n≥3k2−k−2.

The sandpile model on the complete split graph, Motzkin words, and tiered parking functions

We classify recurrent states of the Abelian sandpile model (ASM) on the complete split graph. There are two distinct cases to be considered that depend upon the location of the sink vertex in the complete split graph. This characterisation of decreasing recurrent states is in terms of Motzkin words and can also be characterised in terms of combinatorial necklaces. We also give a characterisation of the recurrent states in terms of a new type of parking function that we call a tiered parking function. These parking functions are characterised by assigning a tier (or colour) to each of the cars, and specifying how many cars of a lower-tier one wishes to have parked before them. We also enumerate the different sets of recurrent configurations studied in this paper, and in doing so derive a formula for the number of spanning trees of the complete split graph that uses a bijective Prüfer code argument.

On (distance) signless Laplacian spectra of graphs

Let Q(G), $${{\mathcal {D}}(G)}$$ and $${{\mathcal {D}}}^Q(G)={{\mathcal {D}}iag(Tr)} + {{\mathcal {D}}(G)}$$ be, respectively, the signless Laplacian matrix, the distance matrix and the distance signless Laplacian matrix of graph G, where $${{\mathcal {D}}iag(Tr)}$$ denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of Q(G) and $${{\mathcal {D}}}^Q(G)$$ will be denoted by $$q_{1} \ge q_{2} \ge \cdots \ge q_{n-1} \ge q_n$$ and $$\partial ^Q_1 \ge \partial ^Q_2 \ge \cdots \ge \partial ^Q_{n-1} \ge \partial ^Q_n$$ , respectively. A graph G which does not share its distance signless Laplacian spectrum with any other non-isomorphic graphs is said to be determined by its distance signless Laplacian spectrum. Characterizing graphs with respect to spectra of graph matrices is challenging. In literature, there are many graphs that are proved to be determined by the spectra of some graph matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix, distance matrix etc.). But there are much fewer graphs that are proved to be determined by the distance signless Laplacian spectrum. Namely, the path graph, the cycle graph, the complement of the path and the complement of the cycle are proved to be determined by the distance signless Laplacian spectra. In this paper, we establish Nordhaus–Gaddum-type results for the least signless Laplacian eigenvalue of graph G. Moreover, we prove that the join graph $$G\vee K_{q}$$ is determined by the distance singless Laplacian spectrum when G is a $$p-2$$ regular graph of order p. Finally, we show that the short kite graph and the complete split graph are determined by the distance signless Laplacian spectra. Our approach for characterizing these graphs with respect to distance signless Laplacian spectra is different from those given in literature.

Structural domination and coloring of some ([formula omitted])-free graphs

We show that every connected induced subgraph of a graph G is dominated by an induced connected split graph if and only if G is C-free, where C is a set of six graphs which includes P7 and C7, and each containing an induced P5. A similar characterization is shown for the class of graphs which are dominated by an induced connected complete split graph. Motivated by these results, we study structural descriptions of some classes of (P7, C7)-free graphs. In particular, we give structural descriptions for the class of (P7, C7, C4, gem)-free graphs and for the class of (P7, C7, C4, diamond)-free graphs. Using these results, we show that every (P7, C7, C4, gem)-free graph G satisfies χ(G)≤2ω(G)−1, and that every (P7, C7, C4, diamond)-free graph H satisfies χ(H)≤max{3,ω(H)}.