Two cycles are said to be intersecting if they share at least one common vertex. Let \(\chi'(G)\) and \(\chi”(G)\) denote the list edge chromatic number and list total chromatic number of a graph \(G\), respectively.In this paper, we proved that for any toroidal graph G without intersecting triangles, \(\chi'(G) \leq \Delta(G) +1\) and \(\chi”(G) \leq \Delta(G)+2\) if \(\Delta(G) \geq 6\), and \(\chi'(G) = \Delta(G)\) if \(\Delta(G) \geq 8\).

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.