Intereting Posts

Possibilities for a group $G$ that acts faithfully on a set of objects with two orbits?
Mandelbrot set: periodicity of secondary and subsequent bulbs as multiples of their parent bulbs
Probability of a random $n \times n$ matrix over $\mathbb F_2$ being nonsingular
Interesting problem of finding surface area of part of a sphere.
Are mini-Mandelbrots known to be found in any fractals other than the Mandelbrot set itself?
Fitting data to a portion of an ellipse or conic section
What is the average length of 2 points on a circle, with generalizations
Determining the action of the operator $D\left(z, \frac d{dz}\right)$
Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?
Tarski Monster group with prime $3$ or $5$
Is there a primitive recursive function which gives the nth digit of $\pi$, despite the table-maker's dilemma?
Every closed convex cone in $ \mathbb{R}^2 $ is polyhedral
Series rearrangement and Riemann's theorem
How can I show using the $\epsilon-\delta$ definition of limit that $u(x,y_0)\to l$ as $x\to x_0?$
Prove that the set of all algebraic numbers is countable

The incidence matrix of a graph is a way to represent the graph. Why go through the trouble of creating this representation of a graph? In other words what are the applications of the incidence matrix or some interesting properties it reveals about its graph?

- How many turns can a chess game take at maximum?
- Binomial Congruence
- Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
- Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s
- Possible divisors of $s(2s+1)$
- number of pairs formed from $2n$ people sitting in a circle
- Minimisation of a distance sum
- Generating function for binomial coefficients $\binom{2n+k}{n}$ with fixed $k$
- How do I calculate these sum-of-sum expressions in terms of the generalized harmonic number?
- Fraction of ordered sequences among all sequences

There are many advantages, especially if the total number of edges is $|E| = \Omega(|V|^2)$. First of all, worst-case constant time for adding, deleting edges, also testing if edge exists (adjacency lists/sets might have some additional $\log n$ factors). Second, simplicity: no advanced structures needed, easy to work with, etc. Moreover some algorithms like to store data for each edge (like flows), matrix representation is then very convenient, and sometimes has nice properties like:

- if $A$ contains $1$ for edges and $0$ otherwise, then $A^k$ contains the number of paths of length $k$ between all vertices,
- if $A$ contains weights (with $\infty$ meaning no edge), then $A^k$ using the min-tropical semiring gives you the lightest paths of length $k$ between all the pairs.

Finally, there is spectral graph theory.

I hope this explains something ðŸ˜‰

Because then one may apply matrix theoretical tools to graph theory problems. One area where it is useful is when you consider flows on a graph, e.g. the flow of current on an electrical circuit and the associated potentials.

Another example is the beautiful Matrix Tree Theorem, which says that the number of spanning trees of a graph is equal to a minor of the Laplacian of the graph, which is a matrix closely related to the incidence matrix.

- Sum of random variable
- Prove that if fewer than $n$ students in class are initially infected, the whole class will never be completely infected.
- Local-Global Principle and the Cassels statement.
- How many isomorphisms are there from $\Bbb Z_{12}$ to $\Bbb Z_{4} \oplus \Bbb Z_{3}$?
- Trust region sub-problem with Jacobi Condition
- what are first and second order logics?
- On the integral $\int_{-\infty}^\infty e^{-(x-ti)^2} dx$
- How to prove that this sequence converges? $\sum_{n=1}^{\infty} \frac{1}{n\ln^2(n)}$
- Continuum between addition, multiplication and exponentiation?
- Prove this proposition concerning a theory with âˆ€âˆƒ-axiomatization
- Example of a subset of $\mathbb{R}^2$ that is closed under vector addition, but not closed under scalar multiplication?
- Evaluating limits at positive and negative infinity
- Prove $H$ is normal subgroup
- Quadratic Formula in Complex Variables
- Find min natural number $n$ so that $2^{2002}$ divides $2001^{n}-1$