Kite Graph Research Articles

Maximum principal ratio of the signless Laplacian of graphs

Let G be a connected graph and Q(G) be the signless Laplacian of G. The principal ratio γ(G) of Q(G) is the ratio of the maximum and minimum entries of the Perron vector of Q(G). In this paper, we consider the maximum principal ratio γ(G) among all connected graphs of order n, and show that for sufficiently large n the extremal graph is a kite graph obtained by identifying an end vertex of a path to any vertex of a complete graph.

The maximum principal ratio of graphs

Let G be a connected graph. The principal ratio of G is the ratio of the maximum and minimum entries of its Perron eigenvector. In 2007, Cioabă and Gregory conjectured that among all connected graphs on n vertices, the kite graph attains the maximum principal ratio. In 2018, Tait and Tobin confirmed the conjecture for sufficiently large n. In this article, we show the conjecture is true for all .

$Kite_{p+2,p}$ is determined by its Laplacian spectrum

$Kite_{n,p}$ denotes the kite graph that is obtained by appending complete graph with order $pgeq4$ to an endpoint of path graph with order $n-p$‎. ‎It is shown that $Kite_{n,p}$ is determined by its adjacency spectrum for all $p$ and $n$ [H‎. ‎Topcu and ‎S‎. ‎Sorgun‎, ‎The kite graph is determined by its adjacency spectrum‎, ‎Applied Math‎. ‎and Comp.‎, ‎330 (2018) 134--142]‎. ‎For $n-p=1$‎, ‎it is proven that $Kite_{n,p}$ is determined by its signless Laplacian spectrum when $ngeq4$‎, ‎$nneq5$ and is also determined by its distance spectrum when $ngeq4$ [K‎. ‎C‎. ‎Das and ‎M‎. ‎Liu‎, ‎Kite graphs are determined by their spectra‎, ‎Applied Math‎. ‎and Comp.‎, ‎297 (2017) 74--78]‎. ‎In this note‎, ‎we say that $Kite_{n,p}$ is determined by its Laplacian spectrum for $n-pleq2$‎.

Maximal distance spectral radius of 4-chromatic planar graphs

We show that the kite graph K4(n) uniquely maximizes the distance spectral radius among all connected 4-chromatic planar graphs on n vertices.

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.

Quadratic embedding constants of squid graph and kite graph

The connected graph G = (V, E) is classified as a Quadratic Embedding (QE) class based on the distance matrix D which is conditionally definite negative and if the quadratic embedding constant (QEC) of the graph G is non-positive. In this study, QEC be calculated for Squid Graph containing C 3, C 4, C 5, and kite graph containing C 4 . All of these graphs are included in the QE class, with QEC = 0 for squid graphs and kite graphs containing C 4, QEC (Sq ( 3,m )) < 0 for squid graphs containing C 3, and also for squid graphs containing C 5.

The chromatic number of local antimagic total edge coloring of some related cycle graphs

A graph G = (V, E) be connected, finite, and undirected graph without multiple edges and loops. Graph G(V, E) consists two sets of vertices V and edge E. A graph called related cycle if the subgraph of graph G contains a cycle. Local antimagic total edge labeling is defined a bijection f : V ∪ E → {1, 2, 3, …, |V | + |E|}, if for any two adjacent edges e1 and e2, wt(e1) ≠ wt(e2), where for e = ab ∈ G, wt(e) = f(a) + f(ab) + f(b). Thus, the local antimagic total edge labeling induces a proper edge coloring of G if each edge e is assigned the color wt(e). The local antimagic total edge chromatic number of G denoted by γlate(G), is the minimum of colors needed to coloring the edges of a graph, the number of distinct induced edge labels over all local antimagic total labeling of G. In this paper we study the local antimagic total edge coloring of some related cycle of graphs and determined the chromatic number of some related cycle graphs namely triangular book Btr, prism graph Pr, sun graph Mr and kite graph Kr, s.

K-Graceful, Odd-Even Graceful, Heronian Mean and Analytic Mean Labeling for the Extended Duplicate Graph of Kite Graph

In this paper, we investigate the extended duplicate graph of kite graph KI3,m admits K-graceful, Odd-Even graceful, Heronian mean and Analytic mean labeling.

The maximum relaxation time of a random walk

We show the minimum spectral gap of the normalized Laplacian over all simple, connected graphs on n vertices is (1+o(1))54n3. This minimum is achieved asymptotically by a double kite graph. Consequently, this leads to sharp upper bounds for the maximum relaxation time of a random walk, settling a conjecture of Aldous and Fill. We also improve an eigenvalue–diameter inequality by giving a new lower bound for the spectral gap of the normalized Laplacian. This eigenvalue lower bound is asymptotically best possible.

Total edge irregularity strength of (n,t)-kite graph

Let G(V, E) be a simple, connected, and undirected graph with vertex set V and edge set E. A total k-labeling is a map that carries vertices and edges of a graph G into a set of positive integer labels {1, 2, …, k}. An edge irregular total k-labeling of a graph G is a labeling of vertices and edges of G in such a way that for any different edges e and f, weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The total edge irregularity strength of G, tes(G), is defined as the minimum k for which a graph G has an edge irregular total k-labeling. An (n, t)-kite graph consist of a cycle of length n with a t-edge path (the tail) attached to one vertex of a cycle. In this paper, we investigate the total edge irregularity strength of the (n, t)-kite graph, with n > 3 and t > 1. We obtain the total edge irregularity strength of the (n, t)-kite graph is tes((n, t)-kite) = .

The kite graph is determined by its adjacency spectrum

The kite graph is obtained by appending the complete graph with p vertices to a pendant vertex of the path graph with q vertices. In this paper, we prove that the kite graph is determined by its adjacency spectrum for all p and q.

Characterizing Graphs Of Maximum Principal Ratio

The principal ratio of a connected graph, denoted γ(G), is the ratio of the maximum and minimum entries of its Perron eigenvector. Cioaba and Gregory (2007) conjectured that the graph on n vertices maximizing γ(G) is a kite graph, that is, a complete graph with a pendant path. In this paper, their conjecture is proved

Further results on edge irregularity strength of graphs

&lt;p&gt;A vertex $k$-labelling $\phi:V(G)\longrightarrow \{1,2,\ldots,k\}$ is called irregular $k$-labeling of the graph $G$ if for every two different edges $e$ and $f$, there is $w_{\phi}(e)\neq w_{\phi}(f)$; where the weight of an edge is given by $e=xy\in E(G)$ is $w_{\phi}(xy)=\phi(x)+\phi(y)$. The minimum $k$ for which the graph $G$ has an edge irregular $k$-labelling is called \emph{edge irregularity strength} of $G$, denoted by $es(G)$.\\&lt;br /&gt;In the paper, we determine the exact value of the edge irregularity strength of caterpillars, $n$-star graphs, $(n,t)$-kite graphs, cycle chains and friendship graphs.&lt;/p&gt;

English

English

Kite graphs determined by their spectra

A kite graph Kin, ω is a graph obtained from a clique Kω and a path Pn−ω by adding an edge between a vertex from the clique and an endpoint from the path. In this note, we prove that Kin,n−1 is determined by its signless Laplacian spectrum when n ≠ 5 and n ≥ 4, and Kin,n−1 is also determined by its distance spectrum when n ≥ 4.

A Graph-based approach for Kite recognition

Kites are huge archaeological structures of stone visible from satellite images. Because of their important number and their wide geographical distribution, automatic recognition of these structures on images is an important step towards understanding these enigmatic remnants. This paper presents a complete identification tool relying on a graph representation of the Kites. As Kites are naturally represented by graphs, graph matching methods are thus the main building blocks in the Kite identification process. However, Kite graphs are disconnected geometric graphs for which traditional graph matching methods are useless. To address this issue, we propose a graph similarity measure adapted for Kite graphs. The proposed approach combines graph invariants with a geometric graph edit distance computation leading to an efficient Kite identification process. We analyze the time complexity of the proposed algorithms and conduct extensive experiments both on real and synthetic Kite graph data sets to attest the effectiveness of the approach. We also perform a set of experimentations on other data sets in order to show that the proposed approach is extensible and quite general.

On the spectral characterization of kite graphs

The \textit{Kite graph}, denoted by $Kite_{p,q}$ is obtained by appending a complete graph $K_{p}$ to a pendant vertex of a path $P_{q}$. In this paper, firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t the adjacency matrix. Let $G$ be a graph which is cospectral with $Kite_{p,q}$ and let $w(G)$ be the clique number of $G$. Then, it is shown that $w(G)\geq p-2q+1$. Also, we prove that $Kite_{p,2}$ graphs are determined by their adjacency spectrum.

Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral

It is shown that the study of the imaginary part and of the corresponding dispersion relations of Feynman graph amplitudes within the differential equations method can provide a powerful tool for the solution of the equations, especially in the massive case.The main features of the approach are illustrated by discussing the simple cases of the 1-loop self-mass and of a particular vertex amplitude, and then used for the evaluation of the two-loop massive sunrise and the QED kite graph (the problem studied by Sabry in 1962), up to first order in the (d−4) expansion.

A Lagrange series approach to the spectrum of the Kite

A Lagrange series around adjustable expansion points to compute the eigenvalues of graphs, whose characteristic polynomial is analytically known, is presented. The computations for the kite graph P_nK_m, whose largest eigenvalue was studied by Stevanovic and Hansen [D. Stevanovic and P. Hansen. The minimum spectral radius of graphs with a given clique number. Electronic Journal of Linear Algebra, 17:110–117, 2008.], are illustrated. It is found that the first term in the Lagrange series already leads to a better approximation than previously published bounds.

Square Graceful Graphs

A ( p, q) graph G(V, E) is said to be a square graceful graph if there exists an injection f :V(G)---{0,1,2,3,..., q2 } such that the induced mapping f p : E(G) --- {1, 4, 9,..., q2} defined by f (uv)=| f (u) f (v)| p = ? is a bijection. The function f is called a square labeling of G . In this paper, we prove that the star n K 1, , bistar m n B , , the graph obtained by the subdivision of the edges of the star K1,n , the graph obtained by the subdivision of the central edge of the bistar Bm,n , the generalised crown n C K 3 1, ? , graph 1 P nK m ? (n ? 2) , the comb 1 P K n? , graph ( , ) m n P S , (3, t) kite graph (t ? 2) and the path Pn are square graceful graphs.