A matrix A = (a~j) has the Edmonds--Johnson property if, for each choice of integral vectors dl, d,., b~, b~, the convex hull of the integral solutions of dt~-x~-d2, bt~-Ax~-b.~ is obtained by adding the inequalities cx~_[~], where c is an integral vector and cx~_J holds for each solution of da~_x~_d2, b~_Ax~_b.. We characterize the Edmonds--Johnson property for integral matrices A which satisf...

The WL-rank of a graph Γ is defined to be the rank coherent configuration Γ. We construct new infinite family strictly Deza Cayley graphs for which equal number vertices. from this are divisible design and integral.

Let (R,m) be a Noetherian local ring and let I ⊆ R be an ideal. This paper studies the question of when mI is integrally closed. Particular attention is focused on the case R is a regular local ring and I is a reduced ideal. This question arose through a question posed by Eisenbud and Mazur on the existence of evolutions. Let (R,m) be a Noetherian local ring and let I ⊆ R be an ideal. In this p...

12 Large singularities of analytic functions 113 12.1 The saddle point 13 Multivariate generating functions 128 14 Mellin and other integral transforms 134 15 Functional equations, recurrences, and combinations of methods 137 15.1 Implicit functions, graphical enumeration, and related

Calculating fractal dimension of the graph a function not simple even for real-valued functions. While through this paper, our intention is to provide some initial theories graphs vector-valued In particular, we give fresh attempt estimate Katugampola fractional integral continuous bounded variation defined on closed interval in $$\mathbb {R}.$$ We prove that 1 and so its integral. Further, Höl...

A gain graph is a graph whose edges are labelled invertibly from a group. A weighted gain graph is a gain graph with vertex weights from a semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and tree-expansion polynomials that are Tutte invariants (they satisfy Tutte’s dele...

The k -dimensional Weisfeiler-Leman procedure ( - WL ), which colors -tuples of vertices in rounds based on the neighborhood structure graph, has proven to be immensely fruitful algorithmic study Graph Isomorphism. More generally, it is fundamental importance understanding and exploiting symmetries graphs various settings. Two are -equivalent if produces same final coloring both graphs. 1 -equi...

‎let $d_{n,m}=big[frac{2n+1-sqrt{17+8(m-n)}}{2}big]$ and‎ ‎$e_{n,m}$ be the graph obtained from a path‎ ‎$p_{d_{n,m}+1}=v_0v_1 cdots v_{d_{n,m}}$ by joining each vertex of‎ ‎$k_{n-d_{n,m}-1}$ to $v_{d_{n,m}}$ and $v_{d_{n,m}-1}$‎, ‎and by‎ ‎joining $m-n+1-{n-d_{n,m}choose 2}$ vertices of $k_{n-d_{n,m}-1}$‎ ‎to $v_{d_{n,m}-2}$‎. ‎zhang‎, ‎liu and zhou [on the maximal eccentric‎ ‎connectivity ind...

the order graph of a group $g$, denoted by $gamma^*(g)$, is a graph whose vertices are subgroups of $g$ and two distinct vertices $h$ and $k$ are adjacent if and only if $|h|big{|}|k|$ or $|k|big{|}|h|$. in this paper, we study the connectivity and diameter of this  graph. also we give a relation between the order graph and prime  graph of a group.