Graphs which are derived from the same graph are called homeomorphic graphs or simply homeomorphs. A \(K_4\)-homeomorph denoted by
\(K_4(a,,c,d,e, f)\) is obtained by subdividing the six paths of a complete graph with four vertices into \(a, b, c,d, e, f\) number of segments, respectively.In this paper, we shall study the chromaticity of \(K_4(a, b,c,d,e, f)\) with exactly two non-adjacent paths of length two. We also give a sufficient and necessary condition for all the graphs in this family to be chromatically
unique.

Let G be a graph with diameter d. An antipodal labeling of G is a function f that assigns to each vertex a
non-negative integer (label) such that for any two vertices \(u\) and \(v\), \(|f(u) — f(v)| \geq d — d(u,v)\), where \(d(u, v)\)
is the distance between \(u\) and \(v\). The span of an antipodal labeling f is \(\max{f(u) — f(v) : u,v \in V(G)}\). The
antipodal number for G, denoted by an\((G)\), is the minimum span of an antipodal labeling for \(G\). Let \(C_n\) denote
the cycle on n vertices. Chartrand \(et al\). \([4]\) determined the value of an\((C_n)\) for \(n \equiv 2 \pmod 4\). In this article we
obtain the value of an\((C_n)\) for \(n \equiv 1 \pmod 4\), confirming a conjecture in \([4]\). Moreover, we settle the case \(n \equiv 3 \pmod 4\), and improve the known lower bound and give an upper bound for the case \(n \equiv 0 \pmod 4\).

We classify all embeddings \(\theta\) : \(PG(n,\mathbb{K}) \rightarrow PG(d,\mathbb{F})\), with \(d \geq \frac{n(n+1)}{2}\)
and \(\mathbb{K},\mathbb{F}\) skew fields with \(|\mathbb{K}| > 2\), such that \(\theta\) maps the set of points of each line of \(PG(n, \mathbb{K})\) to a set of coplanar points of \(PG(n, \mathbb{F})\), and such that the image of \(\theta\) generates \(PG(d, \mathbb{F})\). It turns out that \(d = \frac{1}{2}n(n + 3)\) and all examples “essentially” arise from a similar “full” embedding \(\theta’\) : \(PG(n, \mathbb{K}) \rightarrow PG(d, \mathbb{K})\) by identifying \(\mathbb{K}\) with subfields of F and embedding \(PG(d, \mathbb{K})\) into \(PG(d, \mathbb{F})\) by several ordinary field extensions. These “full” embeddings satisfy one more property and are classified in \([5]\). They relate to the quadric Verone-sean of \(PG(n, \mathbb{K})\) in \(PG(d, \mathbb{K})\) and its projections from subspaces of \(PG(n, \mathbb{K})\) generated by sub-Veroneseans (the point sets corresponding to subspaces of \(PG(n, \mathbb{K})\), if \(\mathbb{K}\) is commutative, and to a degenerate analogue of this, if \(\mathbb{K}\) is noncommutative.

Let \(\mathbb{N}\) be the set of all positive integers, and \(\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\). For any \(h \in \mathbb{N}\), a graph \(G = (V, E)\) is said to be \(\mathbb{Z}_h\)-magic if there exists a labeling \(f: E \rightarrow \mathbb{Z}_h \setminus \{0\}\) such that the induced vertex labeling \(f^+: V \rightarrow \mathbb{Z}_h\), defined by \(f^+(v) = \sum_{uv \in E(v)} f(uv)\), is a constant map. The integer-magic spectrum of \(G\) is the set \(\text{JM}(G) = \{h \in \mathbb{N} \mid G \text{ is } \mathbb{Z}_h\text{-magic}\}\). A sun graph is obtained from attaching a path to each pair of adjacent vertices in an \(n\)-cycle. In this paper, we show that the integer-magic spectra of sun graphs are completely determined.

Let \(e: \mathcal{S} \rightarrow \Sigma\) be a full polarized projective embedding of a dense near polygon \(\mathcal{S}\), i.e., for every point \(p\) of \(\mathcal{S}\), the set \(H_p\) of points at non-maximal distance from \(p\) is mapped by \(e\) into a hyperplane \(\Pi_p\) of \(\Sigma\). We show that if every line of \(S\) is incident with precisely three points or if \(\mathcal{S}\) satisfies a certain property (P\(_y\)) then the map \(p \mapsto \Pi_p\) defines a full polarized embedding \(e^*\) (the so-called dual embedding of \(e\)) of \(\mathcal{S}\) into a subspace of the dual \(\Sigma^*\) of \(\Sigma\). This generalizes a result of \([6]\) where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P\(_y\)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding.

Let \(\mathcal{B}(n,k)\) be the set of bicyclic graphs with \(n\) vertices and \(k\) pendant vertices. In this paper, we determine the unique graph with minimal least eigenvalue among all graphs in \(\mathcal{B}(n,k)\). This extremal graph is the same as that on the Laplacian spectral radius as done by Ji-Ming Guo(The Laplacian spectral radius of bicyclic graphsmwith \(n\) vertices and \(k\) pendant vertices, Science China Mathematics, \(53(8)(2010)2135-2142]\). Moreover, the minimal least eigenvalue is a decreasing function on \(k\).

Gnanajothi conjectured that all trees are odd-graceful and verified this conjecture for all trees with order up to \(10\). Since the
conjecture is open now we present a proof to the odd-gracefulness of all lobsters and show a connection between set-ordered odd-graceful labellings and bipartite graceful labellings in a connected graph.

In this article, the lines not meeting a hyperbolic quadric in PG\((3,q)\) are characterized by their intersection properties with points and planes.

By the classical method for obtaining the values of the Riemann zeta-function at even positive integral arguments, we shall give some functional equational proof of some interesting identities and recurrence relations related to the generalized higher-order Euler and Bernoulli numbers attached to a Dirichlet character \(\chi\) with odd conductor \(d\), and shall show an identity between generalized Euler numbers and generalized Bernoulli numbers. Finally, we remark that any weighted short-interval character sums can be expressed as a linear combination of Dirichlet \(L\)-function values at positive integral arguments, via generalized Bernoulli (or Euler) numbers.

A point set \(X\) in the plane is called a k-distance set if there are exactly \(k\) different distances between two distinct points in \(X\). We classify \(11\)-point \(5\)-distance sets.