By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincaré-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is at least 100 years old, the other two results for graphs were found only recently.

Regular boundary value problems on a distance-regular graph associated with Schrödinger operators are analyzed. These problems include the cases in which the boundary has one or two vertices. In each case, the Green matrices are given in terms of two families of orthogonal polynomials, one of them corresponding with the distance polynomials of the distance-regular graphs.

For graphs G and H the Ramsey number r(G, W) is defined to be the least number p such that every 2-coloring (say green and red) of the edges of the complete graph KP contains either a subgraph G with all its edges green, or a subgraph M with all its edges red. The diagonal Ramsey numbers r(G, G) = r(G) for G = K,, are of particular interest. The values r(&) = 6, and r(&) = 18 are well known, bu...

Given V a finite set, a self–adjoint operator on C(V ), K, is called elliptic if it is positive semi–definite and its lowest eigenvalue is simple. Therefore, there exists a unique, up to sign, unitary function ω ∈ C(V ) satisfying K(ω) = λω and then K is named (λ, ω)–elliptic. Clearly, a (λ, ω)–elliptic operator is singular iff λ = 0. Examples of elliptic operators are the so–called Schrödinger...

let g be a (p, q) graph. let f : v (g) → {1, 2, . . . , k} be a map. for each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of g if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

recently, e. m'{a}v{c}ajov'{a} and m. v{s}koviera proved that every bidirected eulerian graph which admits a nowhere zero flow, admits a nowhere zero $4$-flow. this result shows the validity of bouchet's nowhere zero conjecture for eulerian bidirected graphs. in this paper we prove the same theorem in a different terminology and with a short and simple proof. more precisely, we p...

We describe Green’s relations on the partition monoid, two of its submonoids, and the related Jones and Brauer monoids. Key Words: partition monoid, Jones monoid, ideal structure 2000 Mathematics Subject Classi…cation: 20M20 1 The partition monoid and some submonoids The elements of the partition monoid Pn [5, 8, 9] are equivalence classes of graphs on a vertex set n [ n0 = f1; 2; :::ng [ f10; ...

in this paper we define the removable cycle that, if $im$ is a class of graphs, $gin im$, the cycle $c$ in $g$ is called removable if $g-e(c)in im$. the removable cycles in eulerian graphs have been studied. we characterize eulerian graphs which contain two edge-disjoint removable cycles, and the necessary and sufficient conditions for eulerian graph to have removable cycles h...

The forgotten topological index of a molecular graph $G$ is defined as $F(G)=sum_{vin V(G)}d^{3}(v)$, where $d(u)$ denotes the degree of vertex $u$ in $G$. The first through the sixth smallest forgotten indices among all trees, the first through the third smallest forgotten indices among all connected graph with cyclomatic number $gamma=1,2$, the first through<br /...

In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack’s inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting properties of Schrodinger equation are derived. Mathematics Subject Classific...