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A graph G with p vertices and q edges has antimagic labeling if there is a bijection from the edge set of the graph to the label set { 1 , 2 , ... , q } such that p vertices must have distinct vertex sums, where the vertex sums are determined by adding up all the edge labels incident to each vertex v in V (G). Hartsfield and Ringel conjectured that every connected graph is antimagic, except P 2 . In this study, we identified a class of connected graphs that lend credence to this conjecture. In this paper, we proved that the tensor product of a Wheel Graph, Helm Graph, and Flower graph with Star is antimagic. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Zhejiang University (Science Edition) is the property of Journal of Zhejiang University (Science Edition) Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The Ramsey numbers R (T n , W 8) are determined for each tree graph T n of order n ≥ 7 and maximum degree Δ (T n) equal to either n - 4 or n - 5 . These numbers indicate strong support for the conjecture, due to Chen, Zhang and Zhang and to Hafidh and Baskoro, that R (T n , W m) = 2 n - 1 for each tree graph T n of order n ≥ m - 1 with Δ (T n) ≤ n - m + 2 when m ≥ 4 is even. [ABSTRACT FROM AUTHOR]

Two pairs of disjoint bases P 1 = (R 1 , B 1) and P 2 = (R 2 , B 2) of a matroid M are called equivalent if P 1 can be transformed into P 2 by a series of symmetric exchanges. In 1980, White conjectured that such a sequence always exists whenever R 1 ∪ B 1 = R 2 ∪ B 2 . A strengthening of the conjecture was proposed by Hamidoune, stating that the minimum length of an exchange is at most the rank of the matroid. We propose a weighted variant of Hamidoune's conjecture, where the weight of an exchange depends on the weights of the exchanged elements. We prove the conjecture for several matroid classes: strongly base orderable matroids, split matroids, graphic matroids of wheels, and spikes. [ABSTRACT FROM AUTHOR]

A graph's edge labeling involves the allocation of symbols (colors or numbers) to the edges of a graph governed by specific criteria. Such labeling of a graph G with order n and size m is named edge odd graceful if there is a bijective map φ from the set of edges E (G) = { e 1 , ... , e m } to the set { 1 , 3 , ... , 2 m − 1 } in a way that the derived transformation φ ∗ from the vertex-set V (G) = { v 1 , ... , v n } to the set { 0 , 1 , 2 , ... , 2 m − 1 } given by φ ∗ (u) = ∑ u v ∈ E (G) φ (u v) mod (2 m) is injective. Any graph is named edge odd graceful if it permits an edge odd graceful allocation (Solairaju and Chithra). The primary aim of this study is to define and explore the edge odd graceful labeling of five new families of wheel-related graphs. Consequently, necessary and sufficient conditions for these families to be edge odd graceful are provided. [ABSTRACT FROM AUTHOR]

An incidence of a graph G is a pair (v, e) where v is a vertex of G and e is an edge of G incident with v. Two incidences (v, e) and (w, f) of G are adjacent whenever (i) v = w, or (ii) e = f, or (iii) vw = e or f. An incidence p-colouring of G is a mapping from the set of incidences of G to the set of colours {1,..., p} such that every two adjacent incidences receive distinct colours. Incidence colouring has been introduced by Brualdi and Quinn Massey in 1993 and, since then, studied by several authors. In this paper, we introduce and study the strong version of incidence colouring, where incidences adjacent to the same incidence must also get distinct colours. We determine the exact value of -- or upper bounds on -- the strong incidence chromatic number of several classes of graphs, namely cycles, wheel graphs, trees, ladder graphs, square grids and subclasses of Halin graphs. [ABSTRACT FROM AUTHOR]

Restricted detour distance between two vertices and of a graph is the length of a longest u- v path for the induced condition (V (P))=P. The restricted detour polynomial depends on restricted detour distance and is denoted by D*(G,x) and defined by D*(G,x) = ∑{u,v}xD*(u,v) where the summation is taken over all unordered pairs (u,v) of distinct vertices of u and v of V(G). In this paper, we investigate the restricted detour polynomial of the straight chain of wheel graphs φ(WkB) and compute the restricted detour index of φ(WkB).

Consider G to be a simple graph with n vertices and m edges, and L (G) to be a Laplacian matrix with Laplacian eigenvalues of μ 1 , μ 2 , ... , μ n = z e r o . Write S k (G) = ∑ i = 1 k μ i as the sum of the k-largest Laplacian eigenvalues of G, where k ∈ { 1 , 2 , ... , n } . The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer's conjecture states that S k (G) ≤ m + k + 1 2 , where k = 1 , 2 , ... , n . This paper proves Brouwer's conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph W n + 1 , which provides a better approximation for this upper bound using Brouwer's conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of W n + 1 , n ≥ 3 . We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves. [ABSTRACT FROM AUTHOR]

A graph’s edge labeling involves the allocation of symbols (colors or numbers) to the edges of a graph governed by specific criteria. Such labeling of a graph G with order n and size m is named edge odd graceful if there is a bijective map φ from the set of edges E(G)={e1,…,em} to the set {1,3,…,2m−1} in a way that the derived transformation φ∗ from the vertex-set V(G)={v1,…,vn} to the set {0,1,2,…,2m−1} given by φ∗(u)=∑uv∈E(G)φ(uv)mod(2m) is injective. Any graph is named edge odd graceful if it permits an edge odd graceful allocation (Solairaju and Chithra). The primary aim of this study is to define and explore the edge odd graceful labeling of five new families of wheel-related graphs. Consequently, necessary and sufficient conditions for these families to be edge odd graceful are provided.

This study considers the exact wirelength of embedding complete bipartite graphs into wheel related graphs such as wheel graphs, gear graphs, and helm graphs. [ABSTRACT FROM AUTHOR]

A Grundy k-coloring of a graph G is a proper k-coloring of vertices in G using colors {1, 2, · · ·, k} such that for any two colors i and j, i < j, any vertex colored j is adjacent to some vertex colored i. The First-Fit or Grundy chromatic number (or simply Grundy number) of a graph G, denoted by Γ (G), is the largest integer k, such that there exists a Grundy k-coloring for G. It can be easily seen that Γ (G) equals to the maximum number of colors used by the greedy (or First-Fit) coloring of G [10]. In this paper, we obtain the Grundy chromatic number of middle graph of graph G, denoted by M (G), where G be a cycle or sunlet graph or star graph or wheel graph or helm graph. [ABSTRACT FROM AUTHOR]

Let G(V, E) be a graph with order p and size q. A bijection f : V (G) → {0, 1, 2, · · ·, p − 1} is said to be a cube sum labeling if the induced function f∗ : E(G) → N defined by f∗ (uv) = [f(u)]³ + [f(v)]³ is injective. Such a function f is said to be a cube sum labeling and the graph G is a cube sum graph. In this paper we discuss some algebraic properties and evaluate some families of cube sum graph. [ABSTRACT FROM AUTHOR]

AbstractThe permanent of an n × n matrix [Formula: see text] is defined as [Formula: see text] where the sum is taken over all permutations σ of [Formula: see text] The permanental polynomial of M, denoted by [Formula: see text] is [Formula: see text] where In is the identity matrix of order n. Let G be a simple undirected graph on n vertices and its Laplacian and signless Laplacian matrices be L(G) and Q(G) respectively. The permanental polynomials [Formula: see text] and [Formula: see text] are called the Laplacian permanental polynomial and signless Laplacian permanental polynomial of G respectively. A graph G is said to be determined by its (signless) Laplacian permanental polynomial if all the graphs having the same (signless) Laplacian permanental polynomial with G are isomorphic to G. A graph G is said to be combinedly determined by its Laplacian and signless Laplacian permanental polynomials if all the graphs having (i) the same Laplacian permanental polynomial as [Formula: see text] and (ii) the same signless Laplacian permanental polynomial as [Formula: see text] are isomorphic to G. In this article we investigate the determination of some graphs, namely, star, wheel, friendship graphs and a particular kind of caterpillar graph [Formula: see text] (whose all r non-pendant vertices have the same degree n) by their Laplacian and signless Laplacian permanental polynomials. We show that a kind of caterpillar graphs [Formula: see text] (for [Formula: see text]), wheel graph (up to 7 vertices) and friendship graph (up to 7 vertices) are determined by their (signless) Laplacian permanental polynomials. Further we prove that all friendship graphs and wheel graphs are combinedly determined by their Laplacian and signless Laplacian permanental polynomials.

A (0; 1)-labelling of a set is said to be friendly if approximately one half the elements of the set are labelled 0 and one half labelled 1. Let g be a labelling of the edge set of a graph that is induced by a labelling f of the vertex set. If both g and f are friendly then g is said to be a cordial labelling of the graph. We extend this concept to directed graphs and investigate the cordiality of sets of directed graphs. We investigate a specific type of cordiality on digraphs, a restriction of quasigroup-cordiality called (2; 3)-cordiality. A directed graph is (2; 3)-cordial if there is a friendly labelling f of the vertex set which induces a (1;-1; 0)-labelling of the arc set g such that about one third of the arcs are labelled 1, about one third labelled -1 and about one third labelled 0. In particular we determine which tournaments are (2; 3)-cordial, which orientations of the n-wheel are (2; 3)-cordial, and which orientations of the n-fan are (2; 3)-cordial. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a graph of order p and size q having no isolated vertices. A bijection f : V → {1, 2, 3, ..., p} is called a local edge antimagic labeling if for any two adjacent edges e = uv and é = vw of G, we have w(e) 6= w(e 0 ), where the edge weight w(e = uv) = f(u)+f(v) and w(e 0 ) = f(v)+f(w). A graph G is local edge antimagic if G has a local edge antimagic labeling. The local edge antimagic chromatic number χ 0 lea (G) is defined to be the minimum number of colors taken over all colorings of G induced by local edge antimagic labelings of G. In this paper, we determine the local edge antimagic chromatic number for a friendship graph, wheel graph, fan graph, helm graph, flower graph, and closed helm. [ABSTRACT FROM AUTHOR]

Consider G to be a simple graph with n vertices and m edges, and L(G) to be a Laplacian matrix with Laplacian eigenvalues of μ1,μ2,…,μn=zero. Write Sk(G)=∑i=1kμi as the sum of the k-largest Laplacian eigenvalues of G, where k∈{1,2,…,n}. The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer’s conjecture states that Sk(G)≤m+k+12, where k=1,2,…,n. This paper proves Brouwer’s conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph Wn+1, which provides a better approximation for this upper bound using Brouwer’s conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of Wn+1, n≥3. We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves.

In this paper, it is explicitly determined the r-dynamic chromatic number of the direct product of any given path with either a complete graph or a wheel graph. Illustrative examples are shown for each one of the cases that are studied throughout the paper.

The strong edge coloring of a graph G is a proper edge coloring that assigns a different color to any two edges which are at most two edges apart. The minimum number of color classes that contribute to such a proper coloring is said to be the strong chromatic index of G. This paper defines the strong chromatic index for the generalized Jahangir graphs and the generalized Helm graphs. [ABSTRACT FROM AUTHOR]

We introduce Tribonacci cordial labeling as an extension of Fibonacci cordial labeling, a well-known forms of vertex-labelings. A graph that admits Tribonacci cordial labeling is called Tribonacci cordial graph. In this paper we investigate whether some well-known graphs are Tribonacci cordial. [ABSTRACT FROM AUTHOR]

ABSTRAK Pada artikel ini dibahas sifat-sifat hasil kali matriks (mod 2) terkait graf roda, graf pertemanan, dan graf bunga yang grafikal. Beberapa hasil yang diperoleh, A(Wn)A(Wn)(Mod 2) dan A(Wn)A(Sn)(Mod 2) grafikal apabila n=2k+1 dengan Sn merupakan graf bintang. Selanjutnya, diperoleh A(Wn)A(Go)(mod 2) dan A(Wn)A(G0)(mod 2) grafikal untuk semua n=3 dengan G0 adalah subgraf dari Wn dengan degG0v0=0, degG0vl=degWnvl, untuk 1= l = n. Hasil kali matriks (mod 2) yang grafikal juga diperoleh untuk graf pertemanan dan graf bunga dengan komplemen dan subgrafnya masing- masing. Hasil lebih umum diperoleh untuk kondisi sehingga A(G)A(G)(mod 2) grafikal. ABSTRACT In this paper, we discussed the properties of the wheel, flower and friendship graphs for which the matrix product under modulo 2 was graphical. Let Sn be a star graph and G0 be a subgraph of Wn where degG0v0=0, degG0vl=degWnvl, for 1= l = n. We proved the matrix product A(Wn)A(Wo)(mod 2) and A(Wn)A(Sn)(Mod 2) was graphical for n=2k+1 and the matrix product A(Wn)A(Go)(mod 2) and A(Wn)A(G0)(mod 2) was graphical for all n=3. For the next, a graphical matrix product (mod 2) was also obtained for the friendship graph and the flower graph with its complement and subgraph, respectively. As more general results were obtained for conditions such that A(G)A(G)(mod 2) was graphical.

An equitable edge coloring of a graph is a proper edge coloring for which the difference between any two color classes is at most one. The minimum cardinality of G for such coloring is called equitable edge chromatic number. In this article, we determine the theorem on equitable edge coloring for sunlet graph, wheel graph and helm graph. [ABSTRACT FROM AUTHOR]

Chain graphs and threshold graphs are often referred to as extremal graphs, in the context that, they have the largest spectral radius among all the bipartite graphs (former one) and all the connected graphs (latter one). Nesting in the neighborhood of vertices in the above said extremal graphs have gained the attention of various researchers. Motivated by this structure, we generalize and define a new class of graphs named 'partial chain graphs' and study the properties. We also give the expression for rank, determinant and permanent of these graphs, from which permanent and determinants of well-known wheel graphs, fan graphs, and friendship graphs can be derived. [ABSTRACT FROM AUTHOR]

The main goal of the paper is to answer an unsolved problem. A multicone graph is defined to be the join of a clique and a regular graph, and a wheel as the join of a vertex and a cycle. In this study, we present new classes of multicone graphs that are natural generalizations of wheel graphs and we show that they are determined by their adjacency spectra as well as their Laplacian spectra. We also show that the complements of some of these graphs are determined by their adjacency spectra. In addition, we give a necessary and sufficient condition for perfect graphs cospectral with some of the graphs investigated in the paper. Finally, we conclude with two problems for further study.

Let R (H 1 , H 2) denote the Ramsey number for the graphs H 1 , H 2 , and let J k be K k − e. We present algorithms which enumerate all circulant and block-circulant Ramsey graphs for different types of graphs, thereby obtaining several new lower bounds on Ramsey numbers including: 49 ≤ R (K 3 , J 12) , 36 ≤ R (J 4 , K 8) , 43 ≤ R (J 4 , J 10) , 52 ≤ R (K 4 , J 8) , 37 ≤ R (J 5 , J 6) , 43 ≤ R (J 5 , K 6) , 65 ≤ R (J 5 , J 7). We also use a gluing strategy to derive a new upper bound on R (J 5 , J 6). With both strategies combined, we prove the value of two Ramsey numbers: R (J 5 , J 6) = 37 and R (J 5 , J 7) = 65. We also show that the 64-vertex extremal Ramsey graph for R (J 5 , J 7) is unique. Furthermore, our algorithms also allow the establishment of new lower bounds and exact values on Ramsey numbers involving wheel graphs and complete bipartite graphs, including: R (W 7 , W 4) = 21 , R (W 7 , W 7) = 19 , R (K 3 , 4 , K 3 , 4) = 25 , and R (K 3 , 5 , K 3 , 5) = 33. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigated the edge even graceful labeling property of the join of two graphs. A function f is called an edge even graceful labeling of a graph G=(V(G),E(G)) with p=|V(G)| vertices and q=|E(G)| edges if f:E(G)→{2,4,...,2q} is bijective and the induced function f ∗:V(G) →{0,2,4,⋯,2q−2 }, defined as f ∗ ( x ) = ( ∑ xy ∈ E ( G ) f ( xy ) ) mod ( 2 k ) $ f^{\ast }(x) = ({\sum \nolimits }_{xy \in E(G)} f(xy)~)~\mbox{{mod}}~(2k) $ , where k=m a x(p,q), is an injective function. Sufficient conditions for the complete bipartite graph K m,n =m K 1+n K 1 to have an edge even graceful labeling are established. Also, we introduced an edge even graceful labeling of the join of the graph K 1 with the star graph K 1,n , the wheel graph W n and the sunflower graph s f n for all n ∈ ℕ $n \in \mathbb {N}$ . Finally, we proved that the join of the graph K ¯ 2 $\overline {K}_{2}~$ with the star graph K 1,n , the wheel graph W n and the cyclic graph C n are edge even graceful graphs.

A star coloring of a graph G is a proper vertex coloring in which every path on four vertices in G is not bicolored. The star chromatic number Xs (G) of G is the least number of colors needed to star color G. In this paper, we determine the star chromatic number of lexicographic product of complete graph with complete graph Km[Kn], complete graph with wheel graph Km[Wn], complete graph with path Km[Pn] and path with path Pm[Pn]. [ABSTRACT FROM AUTHOR]

Let $f : V(G) → N be a labeling of the vertices of a graph G. Let S(v) denote the sum of labels of the neighbours of the vertex v in G. If v is an isolated vertex of G, then S(v) = 0. A labeling $f$ is lucky if S(u) ≠ S(v) for every pair of adjacent vertices u and v. The lucky number of a graph G, denoted by η(G), is the least positive integer k such that G has a lucky labeling with {1, 2,..., k\} as the set of labels. In this paper we prove that shell graph, bow graph and wheel graph admits Lucky labeling. [ABSTRACT FROM AUTHOR]

The aim of this study is to compute the edge metric dimension of some subdivision of the wheel graphs. In particular, we determine and compare the metric and edge metric dimensions of the graphs obtained after the cycle, spoke and barycentric subdivisions of the wheel graph. Furthermore, some families of graphs have been constructed through subdivision process for which dim (𝒲 𝒞 𝒫 n , 1) < dim e (𝒲 𝒞 𝒫 n , 1) , and also dim (𝒲 𝒞 𝒴 n , 1) = dim e (𝒲 𝒞 𝒴 n , 1) which partially answer a question in [A. Kelenc, N. Tratnik and I. G. Yero, Uniquely identifying the edges of a graph: The edge metric dimension, Discrete Appl. Math.251 (2018) 204–220]. [ABSTRACT FROM AUTHOR]

Topological index is a useful number associated with molecular graph and the number correlate certain physico-chemical properties of chemical compounds. The concept of line graph and subdivision graph for a graph G has found various applications in chemical research. In this paper, we compute the first, second, and third Zagreb coindices, the F-coindex, the first and second multiplicative Zagreb coindices and the hyper Zagreb coindex of the subdivision graph and the line graph of subdivision graph of the wheel graph. [ABSTRACT FROM AUTHOR]

Fault diagnosis of systems is an important area of study in the design and maintenance of multiprocessor systems. In 2005, Lai et al. [12] introduced conditional diagnosability under the assumption that all the neighbors of any processor in a multiprocessor system cannot be faulty at the same time. In this paper, we completely determine the conditional diagnosability of Cayley graphs generated by wheel graphs W G n under the PMC model. [ABSTRACT FROM AUTHOR]

This article aims to present a novel method, namely wheel partition technique, for constructing a new cyclic 12-fold triple system called cyclic triple factorization denoted by CTF(v). We prove the existence of CTF(v) for v = 12n + 10. Then, an arrangement of v x 2(v - 1) triples of CTF(v) is developed using the idea of decomposition of wheel graph into triangles (triples). Moreover, the starter triples algorithms of CTF(v) are formulated to generate all triples. [ABSTRACT FROM AUTHOR]

In this paper, we consider T-colorings of directed graphs. In particular, we consider as a T-set the set Tr = {0, 1, 2, . . ., r−1, r+1, . . .}. Exact values and bounds of the Tr-span of directed graphs whose underlying graph is a wheel graph are presented.

A vertex labeling φ : V(G) → {1, 2,... , k} is called an edge irregular k-labeling of a graph G if all edges in G have unique weights. The weight of an edge is defined as the sum of the labels of its incident vertices. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper we perform a computer based experiment dealing with the edge irregularity strength of complete bipartite graphs. We also present some bounds on this parameter for wheel related graphs. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the edge even graceful labeling property of the join of two graphs. A function f is called an edge even graceful labeling of a graph G=(V(G),E(G)) with p=|V(G)| vertices and q=|E(G)| edges if f:E(G)→{2,4,...,2q} is bijective and the induced function f∗:V(G) →{0,2,4,⋯,2q−2 }, defined as f ∗ (x) = (∑ xy ∈ E (G) f (xy)) mod (2 k) , where k=max(p,q), is an injective function. Sufficient conditions for the complete bipartite graph Km,n =mK1+nK1 to have an edge even graceful labeling are established. Also, we introduced an edge even graceful labeling of the join of the graph K1 with the star graph K1,n, the wheel graph Wn and the sunflower graph sfn for all n ∈ ℕ . Finally, we proved that the join of the graph K ¯ 2 with the star graph K1,n, the wheel graph Wn and the cyclic graph Cn are edge even graceful graphs. [ABSTRACT FROM AUTHOR]

Solairaju and Chithra introduced a new type of labeling of a graph with vertices and edges called an edge odd graceful labeling if there is a bijection from the edges of the graph to the set such that, when each vertex is assigned the sum of all edges incident to it , where , the resulting vertex labels are distinct. In this paper we proved necessary and sufficient conditions for some path and cycle related graphs to be edge odd graceful such as: Friendship graphs, Wheel graph, Helm graph, Web graph, Double wheel graph, Gear graph, Fan graph, Double fan graph and Polar grid graph.

For a graph $G$ , let $f(G)$ denote the maximum number of edges in a bipartite subgraph of $G$. Given a fixed graph $H$ and a positive integer $m$ , let $f(m,H)$ denote the minimum possible cardinality of $f(G)$ , as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. Alon et al. ['Maximum cuts and judicious partitions in graphs without short cycles', J. Combin. Theory Ser. B 88 (2003), 329–346] conjectured that, for any fixed graph $H$ , there exists an $\unicode[STIX]{x1D716}(H)>0$ such that $f(m,H)\geq m/2+\unicode[STIX]{x1D6FA}(m^{3/4+\unicode[STIX]{x1D716}})$. We show that, for any wheel graph $W_{2k}$ of $2k$ spokes, there exists $c(k)>0$ such that $f(m,W_{2k})\geq m/2+c(k)m^{(2k-1)/(3k-1)}\log m$. In particular, we confirm the conjecture asymptotically for $W_{4}$ and give general lower bounds for $W_{2k+1}$. [ABSTRACT FROM AUTHOR]

Abstract A class of graphs is χ-bounded if there exists a function f : N → N such that for every graph G in the class and an induced subgraph H of G , if H has no clique of size q + 1 , then the chromatic number of H is less than or equal to f (q). We denote by W n the wheel graph on n + 1 vertices. We show that the class of graphs having no vertex-minor isomorphic to W n is χ -bounded. This generalizes several previous results; χ -boundedness for circle graphs, for graphs having no W 5 vertex-minors, and for graphs having no fan vertex-minors. [ABSTRACT FROM AUTHOR]

Let S ⊆ V (G) and κ G (S) denote the maximum number r of edge-disjoint trees T 1, T 2, ..., T r in G such that V (T i) ∩ V (T j) = S for any i, j ∈ { 1, 2, ..., r } and i ≠ j ⁠. For an integer k with 2 ≤ k ≤ n ⁠, the generalized k-connectivity of a graph G is defined as κ k (G) = min { κ G (S) | S ⊆ V (G) and | S | = k } ⁠. The generalized k -connectivity is a generalization of traditional connectivity. In this paper, we focus on the Cayley graph generated by complete graphs and the Cayley graph generated by wheel graphs, denoted by C T n and W G n ⁠, respectively. We study the generalized 3-connectivity of the two kinds of graphs and show that κ 3 (C T n) = n (n − 1) 2 − 1 and κ 3 (W G n) = 2 n − 3 for n ≥ 3 ⁠. [ABSTRACT FROM AUTHOR]

In this paper, we consider T-colorings of directed graphs. In particular, we consider as a T-set the set Tr = {0, 1, 2, . . ., r-1, r+1, . . .}. Exact values and bounds of the Tr-span of directed graphs whose underlying graph is a wheel graph are presented. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We are concerned with families of graphs in which there is a single root-vertex ofunbounded valence, and in which, however, there is a uniform upper bound for the valences of all the other vertices. Using a result of Zagier, we obtain formulas and recursions for the genus distributions of several such families, including the wheel graphs. We show that the region distribution of a wheel graph is approximately proportional to the sequence of Stirling numbers of the first kind. Stahl has previously obtained such a result for embeddings in surfaces whose genus is relatively near to the maximum genus. Here, we generalize Stahl’s result to the entire genus distributions of wheels. Moreover, we derive the genus distributions for four other graph families that have some similarities to wheels. [ABSTRACT FROM AUTHOR]